fabs fraction 1

Percentage Accurate: 91.6% → 99.4%

Time: 9.1s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.35 \cdot 10^{-83}:\\ \;\;\;\;\left|\frac{1}{\frac{y_m}{x + \left(4 - x \cdot z\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y_m}, \frac{-4 - x}{y_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1.35e-83)
   (fabs (/ 1.0 (/ y_m (+ x (- 4.0 (* x z))))))
   (fabs (fma x (/ z y_m) (/ (- -4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.35e-83) {
		tmp = fabs((1.0 / (y_m / (x + (4.0 - (x * z))))));
	} else {
		tmp = fabs(fma(x, (z / y_m), ((-4.0 - x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.35e-83)
		tmp = abs(Float64(1.0 / Float64(y_m / Float64(x + Float64(4.0 - Float64(x * z))))));
	else
		tmp = abs(fma(x, Float64(z / y_m), Float64(Float64(-4.0 - x) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 1.35e-83], N[Abs[N[(1.0 / N[(y$95$m / N[(x + N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.34999999999999996e-83

   1. Initial program 89.5%

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

      1. associate-*l/ 93.2%

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

      2. sub-div 98.8%

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

      3. clear-num 98.7%

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

      4. associate--l+ 98.7%

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

   3. Applied egg-rr 98.7%

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

   if 1.34999999999999996e-83 < y

   1. Initial program 97.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-83}:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x + \left(4 - x \cdot z\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x + 4}{y_m} - z \cdot \frac{x}{y_m}\right|\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-67}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y_m}\right|\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+294}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (- (/ (+ x 4.0) y_m) (* z (/ x y_m))))))
   (if (<= t_0 5e-67)
     (fabs (/ (- (+ x 4.0) (* x z)) y_m))
     (if (<= t_0 5e+294) t_0 (fabs (/ (+ z -1.0) (/ y_m x)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((((x + 4.0) / y_m) - (z * (x / y_m))));
	double tmp;
	if (t_0 <= 5e-67) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else if (t_0 <= 5e+294) {
		tmp = t_0;
	} else {
		tmp = fabs(((z + -1.0) / (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) :: tmp
    t_0 = abs((((x + 4.0d0) / y_m) - (z * (x / y_m))))
    if (t_0 <= 5d-67) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else if (t_0 <= 5d+294) then
        tmp = t_0
    else
        tmp = abs(((z + (-1.0d0)) / (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 = Math.abs((((x + 4.0) / y_m) - (z * (x / y_m))));
	double tmp;
	if (t_0 <= 5e-67) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else if (t_0 <= 5e+294) {
		tmp = t_0;
	} else {
		tmp = Math.abs(((z + -1.0) / (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((((x + 4.0) / y_m) - (z * (x / y_m))))
	tmp = 0
	if t_0 <= 5e-67:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	elif t_0 <= 5e+294:
		tmp = t_0
	else:
		tmp = math.fabs(((z + -1.0) / (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(z * Float64(x / y_m))))
	tmp = 0.0
	if (t_0 <= 5e-67)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	elseif (t_0 <= 5e+294)
		tmp = t_0;
	else
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y_m / x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((((x + 4.0) / y_m) - (z * (x / y_m))));
	tmp = 0.0;
	if (t_0 <= 5e-67)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	elseif (t_0 <= 5e+294)
		tmp = t_0;
	else
		tmp = abs(((z + -1.0) / (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[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 5e-67], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 5e+294], t$95$0, N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))) < 4.9999999999999999e-67

   1. Initial program 89.8%

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

   2. Taylor expanded in y around 0 100.0%

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

   if 4.9999999999999999e-67 < (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))) < 4.9999999999999999e294

   1. Initial program 99.9%

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

   if 4.9999999999999999e294 < (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 76.8%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in x around inf 98.4%

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

   5. Taylor expanded in y around 0 98.4%

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

      1. sub-neg 98.4%

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

      2. metadata-eval 98.4%

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

      3. *-commutative 98.4%

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

      4. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 5 \cdot 10^{-67}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{elif}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 5 \cdot 10^{+294}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 3: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y_m}\right|\\ t_1 := \left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-111}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+31}:\\ \;\;\;\;\left|x \cdot \frac{z}{y_m}\right|\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+73} \lor \neg \left(x \leq 2.9 \cdot 10^{+149}\right) \land x \leq 4.1 \cdot 10^{+232}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* z (/ x y_m)))))
   (if (<= x -1.1e+107)
     t_0
     (if (<= x -2.6e-70)
       t_1
       (if (<= x 2.3e-111)
         (fabs (/ 4.0 y_m))
         (if (<= x 3.3e+31)
           (fabs (* x (/ z y_m)))
           (if (or (<= x 7.4e+73) (and (not (<= x 2.9e+149)) (<= x 4.1e+232)))
             t_0
             t_1)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double t_1 = fabs((z * (x / y_m)));
	double tmp;
	if (x <= -1.1e+107) {
		tmp = t_0;
	} else if (x <= -2.6e-70) {
		tmp = t_1;
	} else if (x <= 2.3e-111) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 3.3e+31) {
		tmp = fabs((x * (z / y_m)));
	} else if ((x <= 7.4e+73) || (!(x <= 2.9e+149) && (x <= 4.1e+232))) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 = abs((x / y_m))
    t_1 = abs((z * (x / y_m)))
    if (x <= (-1.1d+107)) then
        tmp = t_0
    else if (x <= (-2.6d-70)) then
        tmp = t_1
    else if (x <= 2.3d-111) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 3.3d+31) then
        tmp = abs((x * (z / y_m)))
    else if ((x <= 7.4d+73) .or. (.not. (x <= 2.9d+149)) .and. (x <= 4.1d+232)) then
        tmp = t_0
    else
        tmp = t_1
    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 = Math.abs((x / y_m));
	double t_1 = Math.abs((z * (x / y_m)));
	double tmp;
	if (x <= -1.1e+107) {
		tmp = t_0;
	} else if (x <= -2.6e-70) {
		tmp = t_1;
	} else if (x <= 2.3e-111) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 3.3e+31) {
		tmp = Math.abs((x * (z / y_m)));
	} else if ((x <= 7.4e+73) || (!(x <= 2.9e+149) && (x <= 4.1e+232))) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	t_1 = math.fabs((z * (x / y_m)))
	tmp = 0
	if x <= -1.1e+107:
		tmp = t_0
	elif x <= -2.6e-70:
		tmp = t_1
	elif x <= 2.3e-111:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 3.3e+31:
		tmp = math.fabs((x * (z / y_m)))
	elif (x <= 7.4e+73) or (not (x <= 2.9e+149) and (x <= 4.1e+232)):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -1.1e+107)
		tmp = t_0;
	elseif (x <= -2.6e-70)
		tmp = t_1;
	elseif (x <= 2.3e-111)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 3.3e+31)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif ((x <= 7.4e+73) || (!(x <= 2.9e+149) && (x <= 4.1e+232)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	t_1 = abs((z * (x / y_m)));
	tmp = 0.0;
	if (x <= -1.1e+107)
		tmp = t_0;
	elseif (x <= -2.6e-70)
		tmp = t_1;
	elseif (x <= 2.3e-111)
		tmp = abs((4.0 / y_m));
	elseif (x <= 3.3e+31)
		tmp = abs((x * (z / y_m)));
	elseif ((x <= 7.4e+73) || (~((x <= 2.9e+149)) && (x <= 4.1e+232)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.1e+107], t$95$0, If[LessEqual[x, -2.6e-70], t$95$1, If[LessEqual[x, 2.3e-111], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.3e+31], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 7.4e+73], And[N[Not[LessEqual[x, 2.9e+149]], $MachinePrecision], LessEqual[x, 4.1e+232]]], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.1e107 or 3.29999999999999992e31 < x < 7.39999999999999947e73 or 2.9000000000000002e149 < x < 4.10000000000000002e232

   1. Initial program 88.6%

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

      1. associate-*l/ 87.7%

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

      2. sub-div 94.0%

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

      3. clear-num 93.8%

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

      4. associate--l+ 93.8%

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

   3. Applied egg-rr 93.8%

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

   4. Taylor expanded in z around 0 74.2%

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

   5. Taylor expanded in x around inf 74.4%

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

   if -1.1e107 < x < -2.60000000000000002e-70 or 7.39999999999999947e73 < x < 2.9000000000000002e149 or 4.10000000000000002e232 < x

   1. Initial program 91.8%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in x around inf 90.6%

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

   5. Taylor expanded in z around inf 62.4%

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

      1. *-commutative 62.4%

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

      2. associate-*r/ 77.3%

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

   7. Simplified 77.3%

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

   if -2.60000000000000002e-70 < x < 2.3e-111

   1. Initial program 95.2%

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

   2. Taylor expanded in x around 0 84.7%

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

   if 2.3e-111 < x < 3.29999999999999992e31

   1. Initial program 96.5%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in x around inf 59.7%

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

   5. Taylor expanded in z around inf 57.8%

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

      1. associate-*r/ 54.8%

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

   7. Simplified 54.8%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+107}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-70}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-111}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+31}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+73} \lor \neg \left(x \leq 2.9 \cdot 10^{+149}\right) \land x \leq 4.1 \cdot 10^{+232}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y_m}\right|\\ t_1 := \left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-110}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 7.9 \cdot 10^{+31}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{z}}\right|\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+73} \lor \neg \left(x \leq 9.5 \cdot 10^{+148}\right) \land x \leq 1.85 \cdot 10^{+232}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* z (/ x y_m)))))
   (if (<= x -8.5e+107)
     t_0
     (if (<= x -4.2e-71)
       t_1
       (if (<= x 9.6e-110)
         (fabs (/ 4.0 y_m))
         (if (<= x 7.9e+31)
           (fabs (/ x (/ y_m z)))
           (if (or (<= x 6.5e+73) (and (not (<= x 9.5e+148)) (<= x 1.85e+232)))
             t_0
             t_1)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double t_1 = fabs((z * (x / y_m)));
	double tmp;
	if (x <= -8.5e+107) {
		tmp = t_0;
	} else if (x <= -4.2e-71) {
		tmp = t_1;
	} else if (x <= 9.6e-110) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 7.9e+31) {
		tmp = fabs((x / (y_m / z)));
	} else if ((x <= 6.5e+73) || (!(x <= 9.5e+148) && (x <= 1.85e+232))) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 = abs((x / y_m))
    t_1 = abs((z * (x / y_m)))
    if (x <= (-8.5d+107)) then
        tmp = t_0
    else if (x <= (-4.2d-71)) then
        tmp = t_1
    else if (x <= 9.6d-110) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 7.9d+31) then
        tmp = abs((x / (y_m / z)))
    else if ((x <= 6.5d+73) .or. (.not. (x <= 9.5d+148)) .and. (x <= 1.85d+232)) then
        tmp = t_0
    else
        tmp = t_1
    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 = Math.abs((x / y_m));
	double t_1 = Math.abs((z * (x / y_m)));
	double tmp;
	if (x <= -8.5e+107) {
		tmp = t_0;
	} else if (x <= -4.2e-71) {
		tmp = t_1;
	} else if (x <= 9.6e-110) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 7.9e+31) {
		tmp = Math.abs((x / (y_m / z)));
	} else if ((x <= 6.5e+73) || (!(x <= 9.5e+148) && (x <= 1.85e+232))) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	t_1 = math.fabs((z * (x / y_m)))
	tmp = 0
	if x <= -8.5e+107:
		tmp = t_0
	elif x <= -4.2e-71:
		tmp = t_1
	elif x <= 9.6e-110:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 7.9e+31:
		tmp = math.fabs((x / (y_m / z)))
	elif (x <= 6.5e+73) or (not (x <= 9.5e+148) and (x <= 1.85e+232)):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -8.5e+107)
		tmp = t_0;
	elseif (x <= -4.2e-71)
		tmp = t_1;
	elseif (x <= 9.6e-110)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 7.9e+31)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif ((x <= 6.5e+73) || (!(x <= 9.5e+148) && (x <= 1.85e+232)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	t_1 = abs((z * (x / y_m)));
	tmp = 0.0;
	if (x <= -8.5e+107)
		tmp = t_0;
	elseif (x <= -4.2e-71)
		tmp = t_1;
	elseif (x <= 9.6e-110)
		tmp = abs((4.0 / y_m));
	elseif (x <= 7.9e+31)
		tmp = abs((x / (y_m / z)));
	elseif ((x <= 6.5e+73) || (~((x <= 9.5e+148)) && (x <= 1.85e+232)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -8.5e+107], t$95$0, If[LessEqual[x, -4.2e-71], t$95$1, If[LessEqual[x, 9.6e-110], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 7.9e+31], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 6.5e+73], And[N[Not[LessEqual[x, 9.5e+148]], $MachinePrecision], LessEqual[x, 1.85e+232]]], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.4999999999999999e107 or 7.9000000000000003e31 < x < 6.5000000000000001e73 or 9.5000000000000002e148 < x < 1.84999999999999986e232

   1. Initial program 88.6%

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

      1. associate-*l/ 87.7%

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

      2. sub-div 94.0%

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

      3. clear-num 93.8%

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

      4. associate--l+ 93.8%

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

   3. Applied egg-rr 93.8%

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

   4. Taylor expanded in z around 0 74.2%

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

   5. Taylor expanded in x around inf 74.4%

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

   if -8.4999999999999999e107 < x < -4.2000000000000002e-71 or 6.5000000000000001e73 < x < 9.5000000000000002e148 or 1.84999999999999986e232 < x

   1. Initial program 91.8%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in x around inf 90.6%

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

   5. Taylor expanded in z around inf 62.4%

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

      1. *-commutative 62.4%

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

      2. associate-*r/ 77.3%

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

   7. Simplified 77.3%

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

   if -4.2000000000000002e-71 < x < 9.60000000000000026e-110

   1. Initial program 95.2%

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

   2. Taylor expanded in x around 0 84.7%

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

   if 9.60000000000000026e-110 < x < 7.9000000000000003e31

   1. Initial program 96.5%

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

   2. Taylor expanded in z around inf 57.8%

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

      1. mul-1-neg 57.8%

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

      2. associate-*l/ 54.5%

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

      3. distribute-rgt-neg-out 54.5%

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

   4. Simplified 54.5%

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

      1. add-sqr-sqrt 25.3%

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

      2. sqrt-unprod 25.7%

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

      3. sqr-neg 25.7%

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

      4. sqrt-unprod 28.9%

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

      5. add-sqr-sqrt 54.5%

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

      6. associate-/r/ 54.9%

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

   6. Applied egg-rr 54.9%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+107}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-71}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-110}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 7.9 \cdot 10^{+31}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+73} \lor \neg \left(x \leq 9.5 \cdot 10^{+148}\right) \land x \leq 1.85 \cdot 10^{+232}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 5: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y_m}\right|\\ t_1 := \left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-110}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+35}:\\ \;\;\;\;\left|\frac{x \cdot z}{y_m}\right|\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+73} \lor \neg \left(x \leq 2.6 \cdot 10^{+147}\right) \land x \leq 9.5 \cdot 10^{+231}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* z (/ x y_m)))))
   (if (<= x -3.6e+107)
     t_0
     (if (<= x -2.25e-70)
       t_1
       (if (<= x 9.2e-110)
         (fabs (/ 4.0 y_m))
         (if (<= x 2.1e+35)
           (fabs (/ (* x z) y_m))
           (if (or (<= x 3.4e+73) (and (not (<= x 2.6e+147)) (<= x 9.5e+231)))
             t_0
             t_1)))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double t_1 = fabs((z * (x / y_m)));
	double tmp;
	if (x <= -3.6e+107) {
		tmp = t_0;
	} else if (x <= -2.25e-70) {
		tmp = t_1;
	} else if (x <= 9.2e-110) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 2.1e+35) {
		tmp = fabs(((x * z) / y_m));
	} else if ((x <= 3.4e+73) || (!(x <= 2.6e+147) && (x <= 9.5e+231))) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 = abs((x / y_m))
    t_1 = abs((z * (x / y_m)))
    if (x <= (-3.6d+107)) then
        tmp = t_0
    else if (x <= (-2.25d-70)) then
        tmp = t_1
    else if (x <= 9.2d-110) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 2.1d+35) then
        tmp = abs(((x * z) / y_m))
    else if ((x <= 3.4d+73) .or. (.not. (x <= 2.6d+147)) .and. (x <= 9.5d+231)) then
        tmp = t_0
    else
        tmp = t_1
    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 = Math.abs((x / y_m));
	double t_1 = Math.abs((z * (x / y_m)));
	double tmp;
	if (x <= -3.6e+107) {
		tmp = t_0;
	} else if (x <= -2.25e-70) {
		tmp = t_1;
	} else if (x <= 9.2e-110) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 2.1e+35) {
		tmp = Math.abs(((x * z) / y_m));
	} else if ((x <= 3.4e+73) || (!(x <= 2.6e+147) && (x <= 9.5e+231))) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	t_1 = math.fabs((z * (x / y_m)))
	tmp = 0
	if x <= -3.6e+107:
		tmp = t_0
	elif x <= -2.25e-70:
		tmp = t_1
	elif x <= 9.2e-110:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 2.1e+35:
		tmp = math.fabs(((x * z) / y_m))
	elif (x <= 3.4e+73) or (not (x <= 2.6e+147) and (x <= 9.5e+231)):
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -3.6e+107)
		tmp = t_0;
	elseif (x <= -2.25e-70)
		tmp = t_1;
	elseif (x <= 9.2e-110)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 2.1e+35)
		tmp = abs(Float64(Float64(x * z) / y_m));
	elseif ((x <= 3.4e+73) || (!(x <= 2.6e+147) && (x <= 9.5e+231)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	t_1 = abs((z * (x / y_m)));
	tmp = 0.0;
	if (x <= -3.6e+107)
		tmp = t_0;
	elseif (x <= -2.25e-70)
		tmp = t_1;
	elseif (x <= 9.2e-110)
		tmp = abs((4.0 / y_m));
	elseif (x <= 2.1e+35)
		tmp = abs(((x * z) / y_m));
	elseif ((x <= 3.4e+73) || (~((x <= 2.6e+147)) && (x <= 9.5e+231)))
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.6e+107], t$95$0, If[LessEqual[x, -2.25e-70], t$95$1, If[LessEqual[x, 9.2e-110], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.1e+35], N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 3.4e+73], And[N[Not[LessEqual[x, 2.6e+147]], $MachinePrecision], LessEqual[x, 9.5e+231]]], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.5999999999999998e107 or 2.0999999999999999e35 < x < 3.4000000000000002e73 or 2.5999999999999999e147 < x < 9.5000000000000002e231

   1. Initial program 88.6%

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

      1. associate-*l/ 87.7%

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

      2. sub-div 94.0%

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

      3. clear-num 93.8%

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

      4. associate--l+ 93.8%

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

   3. Applied egg-rr 93.8%

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

   4. Taylor expanded in z around 0 74.2%

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

   5. Taylor expanded in x around inf 74.4%

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

   if -3.5999999999999998e107 < x < -2.25000000000000011e-70 or 3.4000000000000002e73 < x < 2.5999999999999999e147 or 9.5000000000000002e231 < x

   1. Initial program 91.8%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in x around inf 90.6%

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

   5. Taylor expanded in z around inf 62.4%

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

      1. *-commutative 62.4%

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

      2. associate-*r/ 77.3%

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

   7. Simplified 77.3%

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

   if -2.25000000000000011e-70 < x < 9.2000000000000006e-110

   1. Initial program 95.2%

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

   2. Taylor expanded in x around 0 84.7%

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

   if 9.2000000000000006e-110 < x < 2.0999999999999999e35

   1. Initial program 96.5%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in x around inf 59.7%

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

   5. Taylor expanded in z around inf 57.8%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+107}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-70}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-110}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+35}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+73} \lor \neg \left(x \leq 2.6 \cdot 10^{+147}\right) \land x \leq 9.5 \cdot 10^{+231}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 6: 65.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{x}{y_m}\right|\\ t_1 := \left|x \cdot \frac{z}{y_m}\right|\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-110}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+34} \lor \neg \left(x \leq 5.2 \cdot 10^{+73}\right) \land x \leq 1.95 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ x y_m))) (t_1 (fabs (* x (/ z y_m)))))
   (if (<= x -1.8e+110)
     t_0
     (if (<= x -1.1e-76)
       t_1
       (if (<= x 9.2e-110)
         (fabs (/ 4.0 y_m))
         (if (or (<= x 8.4e+34) (and (not (<= x 5.2e+73)) (<= x 1.95e+145)))
           t_1
           t_0))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((x / y_m));
	double t_1 = fabs((x * (z / y_m)));
	double tmp;
	if (x <= -1.8e+110) {
		tmp = t_0;
	} else if (x <= -1.1e-76) {
		tmp = t_1;
	} else if (x <= 9.2e-110) {
		tmp = fabs((4.0 / y_m));
	} else if ((x <= 8.4e+34) || (!(x <= 5.2e+73) && (x <= 1.95e+145))) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((x / y_m))
    t_1 = abs((x * (z / y_m)))
    if (x <= (-1.8d+110)) then
        tmp = t_0
    else if (x <= (-1.1d-76)) then
        tmp = t_1
    else if (x <= 9.2d-110) then
        tmp = abs((4.0d0 / y_m))
    else if ((x <= 8.4d+34) .or. (.not. (x <= 5.2d+73)) .and. (x <= 1.95d+145)) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((x / y_m));
	double t_1 = Math.abs((x * (z / y_m)));
	double tmp;
	if (x <= -1.8e+110) {
		tmp = t_0;
	} else if (x <= -1.1e-76) {
		tmp = t_1;
	} else if (x <= 9.2e-110) {
		tmp = Math.abs((4.0 / y_m));
	} else if ((x <= 8.4e+34) || (!(x <= 5.2e+73) && (x <= 1.95e+145))) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((x / y_m))
	t_1 = math.fabs((x * (z / y_m)))
	tmp = 0
	if x <= -1.8e+110:
		tmp = t_0
	elif x <= -1.1e-76:
		tmp = t_1
	elif x <= 9.2e-110:
		tmp = math.fabs((4.0 / y_m))
	elif (x <= 8.4e+34) or (not (x <= 5.2e+73) and (x <= 1.95e+145)):
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(x / y_m))
	t_1 = abs(Float64(x * Float64(z / y_m)))
	tmp = 0.0
	if (x <= -1.8e+110)
		tmp = t_0;
	elseif (x <= -1.1e-76)
		tmp = t_1;
	elseif (x <= 9.2e-110)
		tmp = abs(Float64(4.0 / y_m));
	elseif ((x <= 8.4e+34) || (!(x <= 5.2e+73) && (x <= 1.95e+145)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((x / y_m));
	t_1 = abs((x * (z / y_m)));
	tmp = 0.0;
	if (x <= -1.8e+110)
		tmp = t_0;
	elseif (x <= -1.1e-76)
		tmp = t_1;
	elseif (x <= 9.2e-110)
		tmp = abs((4.0 / y_m));
	elseif ((x <= 8.4e+34) || (~((x <= 5.2e+73)) && (x <= 1.95e+145)))
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.8e+110], t$95$0, If[LessEqual[x, -1.1e-76], t$95$1, If[LessEqual[x, 9.2e-110], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 8.4e+34], And[N[Not[LessEqual[x, 5.2e+73]], $MachinePrecision], LessEqual[x, 1.95e+145]]], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7999999999999998e110 or 8.4000000000000007e34 < x < 5.2000000000000001e73 or 1.9499999999999999e145 < x

   1. Initial program 88.6%

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

      1. associate-*l/ 83.6%

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

      2. sub-div 90.4%

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

      3. clear-num 90.2%

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

      4. associate--l+ 90.2%

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

   3. Applied egg-rr 90.2%

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

   4. Taylor expanded in z around 0 72.6%

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

   5. Taylor expanded in x around inf 72.7%

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

   if -1.7999999999999998e110 < x < -1.1e-76 or 9.2000000000000006e-110 < x < 8.4000000000000007e34 or 5.2000000000000001e73 < x < 1.9499999999999999e145

   1. Initial program 94.2%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in x around inf 76.4%

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

   5. Taylor expanded in z around inf 64.9%

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

      1. associate-*r/ 64.9%

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

   7. Simplified 64.9%

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

   if -1.1e-76 < x < 9.2000000000000006e-110

   1. Initial program 95.2%

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

   2. Taylor expanded in x around 0 84.7%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+110}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-76}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-110}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+34} \lor \neg \left(x \leq 5.2 \cdot 10^{+73}\right) \land x \leq 1.95 \cdot 10^{+145}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 7: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+210}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y_m} + \frac{-1}{y_m}\right)\right|\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+125}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -8.8e+210)
   (fabs (* x (+ (/ z y_m) (/ -1.0 y_m))))
   (if (<= x 6.2e+125)
     (fabs (/ (- (+ x 4.0) (* x z)) y_m))
     (fabs (/ (+ z -1.0) (/ y_m x))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -8.8e+210) {
		tmp = fabs((x * ((z / y_m) + (-1.0 / y_m))));
	} else if (x <= 6.2e+125) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs(((z + -1.0) / (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) :: tmp
    if (x <= (-8.8d+210)) then
        tmp = abs((x * ((z / y_m) + ((-1.0d0) / y_m))))
    else if (x <= 6.2d+125) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs(((z + (-1.0d0)) / (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 tmp;
	if (x <= -8.8e+210) {
		tmp = Math.abs((x * ((z / y_m) + (-1.0 / y_m))));
	} else if (x <= 6.2e+125) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs(((z + -1.0) / (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -8.8e+210:
		tmp = math.fabs((x * ((z / y_m) + (-1.0 / y_m))))
	elif x <= 6.2e+125:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs(((z + -1.0) / (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -8.8e+210)
		tmp = abs(Float64(x * Float64(Float64(z / y_m) + Float64(-1.0 / y_m))));
	elseif (x <= 6.2e+125)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y_m / x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -8.8e+210)
		tmp = abs((x * ((z / y_m) + (-1.0 / y_m))));
	elseif (x <= 6.2e+125)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs(((z + -1.0) / (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -8.8e+210], N[Abs[N[(x * N[(N[(z / y$95$m), $MachinePrecision] + N[(-1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 6.2e+125], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.79999999999999948e210

   1. Initial program 89.3%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in x around inf 99.9%

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

   if -8.79999999999999948e210 < x < 6.2e125

   1. Initial program 94.1%

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

   2. Taylor expanded in y around 0 99.9%

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

   if 6.2e125 < x

   1. Initial program 86.4%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in x around inf 99.6%

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

   5. Taylor expanded in y around 0 87.4%

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

      1. sub-neg 87.4%

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

      2. metadata-eval 87.4%

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

      3. *-commutative 87.4%

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

      4. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+210}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y} + \frac{-1}{y}\right)\right|\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+125}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 8: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 5 \cdot 10^{-75}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y_m} - \frac{x}{\frac{y_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 5e-75)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (- (/ (+ x 4.0) y_m) (/ x (/ y_m z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5e-75) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 5d-75) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs((((x + 4.0d0) / y_m) - (x / (y_m / z))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5e-75) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 5e-75:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((x + 4.0) / y_m) - (x / (y_m / z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 5e-75)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(x / Float64(y_m / z))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 5e-75)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 5e-75], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999979e-75

   1. Initial program 89.6%

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

   2. Taylor expanded in y around 0 98.8%

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

   if 4.99999999999999979e-75 < y

   1. Initial program 97.9%

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

      1. associate-/r/ 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-75}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 9: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 8 \cdot 10^{-75}:\\ \;\;\;\;\left|\frac{1}{\frac{y_m}{x + \left(4 - x \cdot z\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y_m} - \frac{x}{\frac{y_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 8e-75)
   (fabs (/ 1.0 (/ y_m (+ x (- 4.0 (* x z))))))
   (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 <= 8e-75) {
		tmp = fabs((1.0 / (y_m / (x + (4.0 - (x * z))))));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 8d-75) then
        tmp = abs((1.0d0 / (y_m / (x + (4.0d0 - (x * z))))))
    else
        tmp = abs((((x + 4.0d0) / y_m) - (x / (y_m / z))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 8e-75) {
		tmp = Math.abs((1.0 / (y_m / (x + (4.0 - (x * z))))));
	} else {
		tmp = Math.abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 8e-75:
		tmp = math.fabs((1.0 / (y_m / (x + (4.0 - (x * z))))))
	else:
		tmp = math.fabs((((x + 4.0) / y_m) - (x / (y_m / z))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 8e-75)
		tmp = abs(Float64(1.0 / Float64(y_m / Float64(x + Float64(4.0 - Float64(x * z))))));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(x / Float64(y_m / z))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 8e-75)
		tmp = abs((1.0 / (y_m / (x + (4.0 - (x * z))))));
	else
		tmp = abs((((x + 4.0) / y_m) - (x / (y_m / z))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 8e-75], N[Abs[N[(1.0 / N[(y$95$m / N[(x + N[(4.0 - N[(x * z), $MachinePrecision]), $MachinePrecision]), $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 < 7.9999999999999997e-75

   1. Initial program 89.6%

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

      1. associate-*l/ 93.3%

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

      2. sub-div 98.8%

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

      3. clear-num 98.7%

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

      4. associate--l+ 98.7%

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

   3. Applied egg-rr 98.7%

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

   if 7.9999999999999997e-75 < y

   1. Initial program 97.9%

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

      1. associate-/r/ 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-75}:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x + \left(4 - x \cdot z\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 10: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-72} \lor \neg \left(x \leq 9.6 \cdot 10^{-110}\right):\\ \;\;\;\;\left|\frac{z + -1}{\frac{y_m}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -6.5e-72) (not (<= x 9.6e-110)))
   (fabs (/ (+ z -1.0) (/ y_m x)))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -6.5e-72) || !(x <= 9.6e-110)) {
		tmp = fabs(((z + -1.0) / (y_m / x)));
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-6.5d-72)) .or. (.not. (x <= 9.6d-110))) then
        tmp = abs(((z + (-1.0d0)) / (y_m / x)))
    else
        tmp = abs((((-4.0d0) - x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -6.5e-72) || !(x <= 9.6e-110)) {
		tmp = Math.abs(((z + -1.0) / (y_m / x)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -6.5e-72) or not (x <= 9.6e-110):
		tmp = math.fabs(((z + -1.0) / (y_m / x)))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -6.5e-72) || !(x <= 9.6e-110))
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y_m / x)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -6.5e-72) || ~((x <= 9.6e-110)))
		tmp = abs(((z + -1.0) / (y_m / x)));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -6.5e-72], N[Not[LessEqual[x, 9.6e-110]], $MachinePrecision]], N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.4999999999999997e-72 or 9.60000000000000026e-110 < x

   1. Initial program 91.1%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in x around inf 89.3%

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

   5. Taylor expanded in y around 0 84.1%

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

      1. sub-neg 84.1%

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

      2. metadata-eval 84.1%

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

      3. *-commutative 84.1%

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

      4. associate-/l* 89.3%

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

   7. Simplified 89.3%

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

   if -6.4999999999999997e-72 < x < 9.60000000000000026e-110

   1. Initial program 95.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in z around 0 84.7%

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

      1. associate-*r/ 84.7%

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

      2. distribute-lft-in 84.7%

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

      3. metadata-eval 84.7%

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

      4. neg-mul-1 84.7%

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

      5. sub-neg 84.7%

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

   6. Simplified 84.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-72} \lor \neg \left(x \leq 9.6 \cdot 10^{-110}\right):\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 11: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-76}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y_m} + \frac{-1}{y_m}\right)\right|\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-110}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.1e-76)
   (fabs (* x (+ (/ z y_m) (/ -1.0 y_m))))
   (if (<= x 9.6e-110)
     (fabs (/ (- -4.0 x) y_m))
     (fabs (/ (+ z -1.0) (/ y_m x))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.1e-76) {
		tmp = fabs((x * ((z / y_m) + (-1.0 / y_m))));
	} else if (x <= 9.6e-110) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs(((z + -1.0) / (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) :: tmp
    if (x <= (-1.1d-76)) then
        tmp = abs((x * ((z / y_m) + ((-1.0d0) / y_m))))
    else if (x <= 9.6d-110) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs(((z + (-1.0d0)) / (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 tmp;
	if (x <= -1.1e-76) {
		tmp = Math.abs((x * ((z / y_m) + (-1.0 / y_m))));
	} else if (x <= 9.6e-110) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs(((z + -1.0) / (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -1.1e-76:
		tmp = math.fabs((x * ((z / y_m) + (-1.0 / y_m))))
	elif x <= 9.6e-110:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs(((z + -1.0) / (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -1.1e-76)
		tmp = abs(Float64(x * Float64(Float64(z / y_m) + Float64(-1.0 / y_m))));
	elseif (x <= 9.6e-110)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(Float64(z + -1.0) / Float64(y_m / x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -1.1e-76)
		tmp = abs((x * ((z / y_m) + (-1.0 / y_m))));
	elseif (x <= 9.6e-110)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs(((z + -1.0) / (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -1.1e-76], N[Abs[N[(x * N[(N[(z / y$95$m), $MachinePrecision] + N[(-1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 9.6e-110], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z + -1.0), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1e-76

   1. Initial program 90.7%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in x around inf 92.9%

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

   if -1.1e-76 < x < 9.60000000000000026e-110

   1. Initial program 95.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in z around 0 84.7%

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

      1. associate-*r/ 84.7%

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

      2. distribute-lft-in 84.7%

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

      3. metadata-eval 84.7%

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

      4. neg-mul-1 84.7%

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

      5. sub-neg 84.7%

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

   6. Simplified 84.7%

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

   if 9.60000000000000026e-110 < x

   1. Initial program 91.3%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in x around inf 86.8%

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

   5. Taylor expanded in y around 0 82.9%

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

      1. sub-neg 82.9%

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

      2. metadata-eval 82.9%

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

      3. *-commutative 82.9%

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

      4. associate-/l* 86.8%

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

   7. Simplified 86.8%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-76}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y} + \frac{-1}{y}\right)\right|\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-110}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z + -1}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 12: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+52}:\\ \;\;\;\;\left|x \cdot \frac{z}{y_m}\right|\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+111}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -1.06e+52)
   (fabs (* x (/ z y_m)))
   (if (<= z 3e+111) (fabs (/ (- -4.0 x) y_m)) (fabs (/ x (/ y_m z))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.06e+52) {
		tmp = fabs((x * (z / y_m)));
	} else if (z <= 3e+111) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((x / (y_m / z)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.06d+52)) then
        tmp = abs((x * (z / y_m)))
    else if (z <= 3d+111) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((x / (y_m / z)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.06e+52) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (z <= 3e+111) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((x / (y_m / z)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -1.06e+52:
		tmp = math.fabs((x * (z / y_m)))
	elif z <= 3e+111:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((x / (y_m / z)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -1.06e+52)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (z <= 3e+111)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(x / Float64(y_m / z)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -1.06e+52)
		tmp = abs((x * (z / y_m)));
	elseif (z <= 3e+111)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((x / (y_m / z)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -1.06e+52], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3e+111], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.0599999999999999e52

   1. Initial program 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in x around inf 75.5%

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

   5. Taylor expanded in z around inf 73.8%

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

      1. associate-*r/ 75.5%

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

   7. Simplified 75.5%

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

   if -1.0599999999999999e52 < z < 3e111

   1. Initial program 95.0%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in z around 0 91.1%

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

      1. associate-*r/ 91.1%

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

      2. distribute-lft-in 91.1%

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

      3. metadata-eval 91.1%

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

      4. neg-mul-1 91.1%

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

      5. sub-neg 91.1%

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

   6. Simplified 91.1%

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

   if 3e111 < z

   1. Initial program 82.9%

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

   2. Taylor expanded in z around inf 73.1%

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

      1. mul-1-neg 73.1%

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

      2. associate-*l/ 82.0%

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

      3. distribute-rgt-neg-out 82.0%

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

   4. Simplified 82.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 58.0%

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

      3. sqr-neg 58.0%

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

      4. sqrt-unprod 81.7%

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

      5. add-sqr-sqrt 82.0%

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

      6. associate-/r/ 84.0%

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

   6. Applied egg-rr 84.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{+52}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+111}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 13: 68.3% accurate, 1.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 4.0))) (fabs (/ x y_m)) (fabs (/ 4.0 y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.55) || !(x <= 4.0)) {
		tmp = fabs((x / y_m));
	} else {
		tmp = fabs((4.0 / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.55d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y_m))
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.55) || !(x <= 4.0)) {
		tmp = Math.abs((x / y_m));
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1.55) or not (x <= 4.0):
		tmp = math.fabs((x / y_m))
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 4.0))
		tmp = abs(Float64(x / y_m));
	else
		tmp = abs(Float64(4.0 / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -1.55) || ~((x <= 4.0)))
		tmp = abs((x / y_m));
	else
		tmp = abs((4.0 / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 4 < x

   1. Initial program 89.5%

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

      1. associate-*l/ 85.2%

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

      2. sub-div 92.4%

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

      3. clear-num 92.3%

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

      4. associate--l+ 92.3%

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

   3. Applied egg-rr 92.3%

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

   4. Taylor expanded in z around 0 62.3%

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

   5. Taylor expanded in x around inf 61.3%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0 72.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 14: 40.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \left|\frac{4}{y_m}\right| \end{array} \]

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	return fabs((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 = abs((4.0d0 / y_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.6%

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

2. Taylor expanded in x around 0 39.7%

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

3. Final simplification 39.7%

   \[\leadsto \left|\frac{4}{y}\right| \]

Reproduce

herbie shell --seed 2023335 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))