fabs fraction 1

Percentage Accurate: 91.1% → 99.6%

Time: 6.6s

Alternatives: 11

Speedup: 1.0×

• 20.7% 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: 91.1% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.5 \cdot 10^{-70}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y_m}, \frac{-4 - x}{y_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1.5e-70)
   (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
   (fabs (fma x (/ z y_m) (/ (- -4.0 x) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.5e-70) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = fabs(fma(x, (z / y_m), ((-4.0 - x) / y_m)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.5e-70)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	else
		tmp = abs(fma(x, Float64(z / y_m), Float64(Float64(-4.0 - x) / y_m)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 1.5e-70], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5000000000000001e-70

   1. Initial program 91.5%

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

   2. Taylor expanded in y around 0 96.1%

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

   if 1.5000000000000001e-70 < y

   1. Initial program 98.7%

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

   3. Simplified 99.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-70}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 2: 65.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y_m}\right|\\ t_1 := \left|\frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.95 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-123} \lor \neg \left(x \leq 5.1 \cdot 10^{-124}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ x y_m))))
   (if (<= x -1.7e+207)
     t_0
     (if (<= x -3.9e+129)
       t_1
       (if (<= x -3.2e+72)
         t_0
         (if (<= x -2.95e+16)
           t_1
           (if (or (<= x -6e-123) (not (<= x 5.1e-124)))
             t_0
             (fabs (/ 4.0 y_m)))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z * (x / y_m)));
	double t_1 = fabs((x / y_m));
	double tmp;
	if (x <= -1.7e+207) {
		tmp = t_0;
	} else if (x <= -3.9e+129) {
		tmp = t_1;
	} else if (x <= -3.2e+72) {
		tmp = t_0;
	} else if (x <= -2.95e+16) {
		tmp = t_1;
	} else if ((x <= -6e-123) || !(x <= 5.1e-124)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    t_1 = abs((x / y_m))
    if (x <= (-1.7d+207)) then
        tmp = t_0
    else if (x <= (-3.9d+129)) then
        tmp = t_1
    else if (x <= (-3.2d+72)) then
        tmp = t_0
    else if (x <= (-2.95d+16)) then
        tmp = t_1
    else if ((x <= (-6d-123)) .or. (.not. (x <= 5.1d-124))) then
        tmp = t_0
    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 t_0 = Math.abs((z * (x / y_m)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (x <= -1.7e+207) {
		tmp = t_0;
	} else if (x <= -3.9e+129) {
		tmp = t_1;
	} else if (x <= -3.2e+72) {
		tmp = t_0;
	} else if (x <= -2.95e+16) {
		tmp = t_1;
	} else if ((x <= -6e-123) || !(x <= 5.1e-124)) {
		tmp = t_0;
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z * (x / y_m)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if x <= -1.7e+207:
		tmp = t_0
	elif x <= -3.9e+129:
		tmp = t_1
	elif x <= -3.2e+72:
		tmp = t_0
	elif x <= -2.95e+16:
		tmp = t_1
	elif (x <= -6e-123) or not (x <= 5.1e-124):
		tmp = t_0
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z * Float64(x / y_m)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -1.7e+207)
		tmp = t_0;
	elseif (x <= -3.9e+129)
		tmp = t_1;
	elseif (x <= -3.2e+72)
		tmp = t_0;
	elseif (x <= -2.95e+16)
		tmp = t_1;
	elseif ((x <= -6e-123) || !(x <= 5.1e-124))
		tmp = t_0;
	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)
	t_0 = abs((z * (x / y_m)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -1.7e+207)
		tmp = t_0;
	elseif (x <= -3.9e+129)
		tmp = t_1;
	elseif (x <= -3.2e+72)
		tmp = t_0;
	elseif (x <= -2.95e+16)
		tmp = t_1;
	elseif ((x <= -6e-123) || ~((x <= 5.1e-124)))
		tmp = t_0;
	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_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.7e+207], t$95$0, If[LessEqual[x, -3.9e+129], t$95$1, If[LessEqual[x, -3.2e+72], t$95$0, If[LessEqual[x, -2.95e+16], t$95$1, If[Or[LessEqual[x, -6e-123], N[Not[LessEqual[x, 5.1e-124]], $MachinePrecision]], t$95$0, N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6999999999999999e207 or -3.8999999999999997e129 < x < -3.2000000000000001e72 or -2.95e16 < x < -5.99999999999999968e-123 or 5.1000000000000001e-124 < x

   1. Initial program 92.7%

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

   2. Taylor expanded in z around inf 61.7%

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

      1. mul-1-neg 61.7%

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

      2. associate-*l/ 66.6%

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

      3. distribute-rgt-neg-out 66.6%

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

   4. Simplified 66.6%

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

      1. add-sqr-sqrt 30.9%

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

      2. sqrt-unprod 49.7%

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

      3. sqr-neg 49.7%

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

      4. sqrt-unprod 35.6%

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

      5. add-sqr-sqrt 66.6%

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

      6. expm1-log1p-u 36.5%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y} \cdot z\right)\right)}\right| \]

      7. expm1-udef 28.1%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{y} \cdot z\right)} - 1}\right| \]

      8. associate-*l/ 25.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot z}{y}}\right)} - 1\right| \]

   6. Applied egg-rr 25.3%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)} - 1}\right| \]
   7. Step-by-step derivation

      1. expm1-def 34.2%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)\right)}\right| \]

      2. expm1-log1p 61.7%

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

      3. associate-*l/ 66.6%

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

   8. Simplified 66.6%

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

   if -1.6999999999999999e207 < x < -3.8999999999999997e129 or -3.2000000000000001e72 < x < -2.95e16

   1. Initial program 88.3%

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

   2. Taylor expanded in z around 0 83.1%

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

      1. associate-*r/ 83.1%

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

      2. metadata-eval 83.1%

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

   4. Simplified 83.1%

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

   5. Taylor expanded in x around inf 83.1%

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

   if -5.99999999999999968e-123 < x < 5.1000000000000001e-124

   1. Initial program 97.6%

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

   2. Taylor expanded in x around 0 90.3%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+207}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{+129}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{+72}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.95 \cdot 10^{+16}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-123} \lor \neg \left(x \leq 5.1 \cdot 10^{-124}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 3: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y_m}\right|\\ t_1 := \left|\frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-72}:\\ \;\;\;\;\left|x \cdot \frac{z}{y_m}\right|\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ x y_m))))
   (if (<= x -1.2e+207)
     t_0
     (if (<= x -5.1e+129)
       t_1
       (if (<= x -1.55e+73)
         t_0
         (if (<= x -6.5e+15)
           t_1
           (if (<= x -3.8e-72)
             (fabs (* x (/ z y_m)))
             (if (<= x 4.6e-124) (fabs (/ 4.0 y_m)) t_0))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z * (x / y_m)));
	double t_1 = fabs((x / y_m));
	double tmp;
	if (x <= -1.2e+207) {
		tmp = t_0;
	} else if (x <= -5.1e+129) {
		tmp = t_1;
	} else if (x <= -1.55e+73) {
		tmp = t_0;
	} else if (x <= -6.5e+15) {
		tmp = t_1;
	} else if (x <= -3.8e-72) {
		tmp = fabs((x * (z / y_m)));
	} else if (x <= 4.6e-124) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    t_1 = abs((x / y_m))
    if (x <= (-1.2d+207)) then
        tmp = t_0
    else if (x <= (-5.1d+129)) then
        tmp = t_1
    else if (x <= (-1.55d+73)) then
        tmp = t_0
    else if (x <= (-6.5d+15)) then
        tmp = t_1
    else if (x <= (-3.8d-72)) then
        tmp = abs((x * (z / y_m)))
    else if (x <= 4.6d-124) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((z * (x / y_m)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (x <= -1.2e+207) {
		tmp = t_0;
	} else if (x <= -5.1e+129) {
		tmp = t_1;
	} else if (x <= -1.55e+73) {
		tmp = t_0;
	} else if (x <= -6.5e+15) {
		tmp = t_1;
	} else if (x <= -3.8e-72) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (x <= 4.6e-124) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z * (x / y_m)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if x <= -1.2e+207:
		tmp = t_0
	elif x <= -5.1e+129:
		tmp = t_1
	elif x <= -1.55e+73:
		tmp = t_0
	elif x <= -6.5e+15:
		tmp = t_1
	elif x <= -3.8e-72:
		tmp = math.fabs((x * (z / y_m)))
	elif x <= 4.6e-124:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z * Float64(x / y_m)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -1.2e+207)
		tmp = t_0;
	elseif (x <= -5.1e+129)
		tmp = t_1;
	elseif (x <= -1.55e+73)
		tmp = t_0;
	elseif (x <= -6.5e+15)
		tmp = t_1;
	elseif (x <= -3.8e-72)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (x <= 4.6e-124)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((z * (x / y_m)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -1.2e+207)
		tmp = t_0;
	elseif (x <= -5.1e+129)
		tmp = t_1;
	elseif (x <= -1.55e+73)
		tmp = t_0;
	elseif (x <= -6.5e+15)
		tmp = t_1;
	elseif (x <= -3.8e-72)
		tmp = abs((x * (z / y_m)));
	elseif (x <= 4.6e-124)
		tmp = abs((4.0 / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.2e+207], t$95$0, If[LessEqual[x, -5.1e+129], t$95$1, If[LessEqual[x, -1.55e+73], t$95$0, If[LessEqual[x, -6.5e+15], t$95$1, If[LessEqual[x, -3.8e-72], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.6e-124], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.2e207 or -5.09999999999999996e129 < x < -1.55e73 or 4.60000000000000024e-124 < x

   1. Initial program 91.6%

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

   2. Taylor expanded in z around inf 61.0%

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

      1. mul-1-neg 61.0%

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

      2. associate-*l/ 67.9%

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

      3. distribute-rgt-neg-out 67.9%

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

   4. Simplified 67.9%

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

      1. add-sqr-sqrt 32.7%

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

      2. sqrt-unprod 52.9%

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

      3. sqr-neg 52.9%

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

      4. sqrt-unprod 35.1%

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

      5. add-sqr-sqrt 67.9%

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

      6. expm1-log1p-u 36.9%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y} \cdot z\right)\right)}\right| \]

      7. expm1-udef 29.1%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{y} \cdot z\right)} - 1}\right| \]

      8. associate-*l/ 25.5%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot z}{y}}\right)} - 1\right| \]

   6. Applied egg-rr 25.5%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)} - 1}\right| \]
   7. Step-by-step derivation

      1. expm1-def 33.3%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)\right)}\right| \]

      2. expm1-log1p 61.0%

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

      3. associate-*l/ 67.9%

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

   8. Simplified 67.9%

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

   if -1.2e207 < x < -5.09999999999999996e129 or -1.55e73 < x < -6.5e15

   1. Initial program 88.3%

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

   2. Taylor expanded in z around 0 83.1%

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

      1. associate-*r/ 83.1%

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

      2. metadata-eval 83.1%

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

   4. Simplified 83.1%

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

   5. Taylor expanded in x around inf 83.1%

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

   if -6.5e15 < x < -3.80000000000000002e-72

   1. Initial program 95.0%

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

   2. Taylor expanded in z around inf 68.6%

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

      1. mul-1-neg 68.6%

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

      2. associate-*l/ 64.0%

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

      3. distribute-rgt-neg-out 64.0%

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

   4. Simplified 64.0%

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

      1. add-sqr-sqrt 21.1%

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

      2. sqrt-unprod 40.4%

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

      3. sqr-neg 40.4%

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

      4. sqrt-unprod 42.6%

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

      5. add-sqr-sqrt 64.0%

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

      6. associate-*l/ 68.6%

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

      7. associate-/l* 68.6%

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

   6. Applied egg-rr 68.6%

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

      1. clear-num 68.6%

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

      2. associate-/r/ 68.8%

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

      3. clear-num 68.8%

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

   8. Applied egg-rr 68.8%

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

   if -3.80000000000000002e-72 < x < 4.60000000000000024e-124

   1. Initial program 97.9%

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

   2. Taylor expanded in x around 0 85.2%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+207}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{+129}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{+73}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-72}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 66.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y_m}\right|\\ t_1 := \left|\frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -2 \cdot 10^{+207}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{z}}\right|\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6600000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-72}:\\ \;\;\;\;\left|x \cdot \frac{z}{y_m}\right|\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ x y_m))))
   (if (<= x -2e+207)
     (fabs (/ x (/ y_m z)))
     (if (<= x -3.3e+129)
       t_1
       (if (<= x -5e+72)
         t_0
         (if (<= x -6600000000000.0)
           t_1
           (if (<= x -1e-72)
             (fabs (* x (/ z y_m)))
             (if (<= x 5.1e-124) (fabs (/ 4.0 y_m)) t_0))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z * (x / y_m)));
	double t_1 = fabs((x / y_m));
	double tmp;
	if (x <= -2e+207) {
		tmp = fabs((x / (y_m / z)));
	} else if (x <= -3.3e+129) {
		tmp = t_1;
	} else if (x <= -5e+72) {
		tmp = t_0;
	} else if (x <= -6600000000000.0) {
		tmp = t_1;
	} else if (x <= -1e-72) {
		tmp = fabs((x * (z / y_m)));
	} else if (x <= 5.1e-124) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    t_1 = abs((x / y_m))
    if (x <= (-2d+207)) then
        tmp = abs((x / (y_m / z)))
    else if (x <= (-3.3d+129)) then
        tmp = t_1
    else if (x <= (-5d+72)) then
        tmp = t_0
    else if (x <= (-6600000000000.0d0)) then
        tmp = t_1
    else if (x <= (-1d-72)) then
        tmp = abs((x * (z / y_m)))
    else if (x <= 5.1d-124) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((z * (x / y_m)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (x <= -2e+207) {
		tmp = Math.abs((x / (y_m / z)));
	} else if (x <= -3.3e+129) {
		tmp = t_1;
	} else if (x <= -5e+72) {
		tmp = t_0;
	} else if (x <= -6600000000000.0) {
		tmp = t_1;
	} else if (x <= -1e-72) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (x <= 5.1e-124) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z * (x / y_m)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if x <= -2e+207:
		tmp = math.fabs((x / (y_m / z)))
	elif x <= -3.3e+129:
		tmp = t_1
	elif x <= -5e+72:
		tmp = t_0
	elif x <= -6600000000000.0:
		tmp = t_1
	elif x <= -1e-72:
		tmp = math.fabs((x * (z / y_m)))
	elif x <= 5.1e-124:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z * Float64(x / y_m)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -2e+207)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif (x <= -3.3e+129)
		tmp = t_1;
	elseif (x <= -5e+72)
		tmp = t_0;
	elseif (x <= -6600000000000.0)
		tmp = t_1;
	elseif (x <= -1e-72)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (x <= 5.1e-124)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((z * (x / y_m)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -2e+207)
		tmp = abs((x / (y_m / z)));
	elseif (x <= -3.3e+129)
		tmp = t_1;
	elseif (x <= -5e+72)
		tmp = t_0;
	elseif (x <= -6600000000000.0)
		tmp = t_1;
	elseif (x <= -1e-72)
		tmp = abs((x * (z / y_m)));
	elseif (x <= 5.1e-124)
		tmp = abs((4.0 / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2e+207], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -3.3e+129], t$95$1, If[LessEqual[x, -5e+72], t$95$0, If[LessEqual[x, -6600000000000.0], t$95$1, If[LessEqual[x, -1e-72], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5.1e-124], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.0000000000000001e207

   1. Initial program 87.8%

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

   2. Taylor expanded in z around inf 62.2%

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

      1. mul-1-neg 62.2%

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

      2. associate-*l/ 77.3%

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

      3. distribute-rgt-neg-out 77.3%

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

   4. Simplified 77.3%

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

      1. add-sqr-sqrt 36.8%

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

      2. sqrt-unprod 69.8%

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

      3. sqr-neg 69.8%

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

      4. sqrt-unprod 40.5%

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

      5. add-sqr-sqrt 77.3%

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

      6. associate-*l/ 62.2%

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

      7. associate-/l* 77.4%

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

   6. Applied egg-rr 77.4%

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

   if -2.0000000000000001e207 < x < -3.2999999999999999e129 or -4.99999999999999992e72 < x < -6.6e12

   1. Initial program 88.3%

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

   2. Taylor expanded in z around 0 83.1%

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

      1. associate-*r/ 83.1%

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

      2. metadata-eval 83.1%

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

   4. Simplified 83.1%

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

   5. Taylor expanded in x around inf 83.1%

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

   if -3.2999999999999999e129 < x < -4.99999999999999992e72 or 5.1000000000000001e-124 < x

   1. Initial program 92.6%

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

   2. Taylor expanded in z around inf 60.7%

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

      1. mul-1-neg 60.7%

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

      2. associate-*l/ 65.5%

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

      3. distribute-rgt-neg-out 65.5%

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

   4. Simplified 65.5%

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

      1. add-sqr-sqrt 31.6%

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

      2. sqrt-unprod 48.6%

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

      3. sqr-neg 48.6%

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

      4. sqrt-unprod 33.8%

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

      5. add-sqr-sqrt 65.5%

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

      6. expm1-log1p-u 36.2%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y} \cdot z\right)\right)}\right| \]

      7. expm1-udef 26.5%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{y} \cdot z\right)} - 1}\right| \]

      8. associate-*l/ 22.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot z}{y}}\right)} - 1\right| \]

   6. Applied egg-rr 22.9%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)} - 1}\right| \]
   7. Step-by-step derivation

      1. expm1-def 32.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)\right)}\right| \]

      2. expm1-log1p 60.7%

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

      3. associate-*l/ 65.5%

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

   8. Simplified 65.5%

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

   if -6.6e12 < x < -9.9999999999999997e-73

   1. Initial program 95.0%

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

   2. Taylor expanded in z around inf 68.6%

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

      1. mul-1-neg 68.6%

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

      2. associate-*l/ 64.0%

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

      3. distribute-rgt-neg-out 64.0%

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

   4. Simplified 64.0%

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

      1. add-sqr-sqrt 21.1%

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

      2. sqrt-unprod 40.4%

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

      3. sqr-neg 40.4%

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

      4. sqrt-unprod 42.6%

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

      5. add-sqr-sqrt 64.0%

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

      6. associate-*l/ 68.6%

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

      7. associate-/l* 68.6%

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

   6. Applied egg-rr 68.6%

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

      1. clear-num 68.6%

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

      2. associate-/r/ 68.8%

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

      3. clear-num 68.8%

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

   8. Applied egg-rr 68.8%

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

   if -9.9999999999999997e-73 < x < 5.1000000000000001e-124

   1. Initial program 97.9%

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

   2. Taylor expanded in x around 0 85.2%

      \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+207}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{+129}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+72}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -6600000000000:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-72}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 5: 66.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{z}{\frac{y_m}{x}}\right|\\ t_1 := \left|\frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{+72}:\\ \;\;\;\;\left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-73}:\\ \;\;\;\;\left|x \cdot \frac{z}{y_m}\right|\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ z (/ y_m x)))) (t_1 (fabs (/ x y_m))))
   (if (<= x -1.05e+207)
     t_0
     (if (<= x -5.2e+129)
       t_1
       (if (<= x -2.6e+72)
         (fabs (* z (/ x y_m)))
         (if (<= x -7.2e+15)
           t_1
           (if (<= x -8e-73)
             (fabs (* x (/ z y_m)))
             (if (<= x 4.8e-124) (fabs (/ 4.0 y_m)) t_0))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z / (y_m / x)));
	double t_1 = fabs((x / y_m));
	double tmp;
	if (x <= -1.05e+207) {
		tmp = t_0;
	} else if (x <= -5.2e+129) {
		tmp = t_1;
	} else if (x <= -2.6e+72) {
		tmp = fabs((z * (x / y_m)));
	} else if (x <= -7.2e+15) {
		tmp = t_1;
	} else if (x <= -8e-73) {
		tmp = fabs((x * (z / y_m)));
	} else if (x <= 4.8e-124) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z / (y_m / x)))
    t_1 = abs((x / y_m))
    if (x <= (-1.05d+207)) then
        tmp = t_0
    else if (x <= (-5.2d+129)) then
        tmp = t_1
    else if (x <= (-2.6d+72)) then
        tmp = abs((z * (x / y_m)))
    else if (x <= (-7.2d+15)) then
        tmp = t_1
    else if (x <= (-8d-73)) then
        tmp = abs((x * (z / y_m)))
    else if (x <= 4.8d-124) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((z / (y_m / x)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (x <= -1.05e+207) {
		tmp = t_0;
	} else if (x <= -5.2e+129) {
		tmp = t_1;
	} else if (x <= -2.6e+72) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (x <= -7.2e+15) {
		tmp = t_1;
	} else if (x <= -8e-73) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (x <= 4.8e-124) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z / (y_m / x)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if x <= -1.05e+207:
		tmp = t_0
	elif x <= -5.2e+129:
		tmp = t_1
	elif x <= -2.6e+72:
		tmp = math.fabs((z * (x / y_m)))
	elif x <= -7.2e+15:
		tmp = t_1
	elif x <= -8e-73:
		tmp = math.fabs((x * (z / y_m)))
	elif x <= 4.8e-124:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z / Float64(y_m / x)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -1.05e+207)
		tmp = t_0;
	elseif (x <= -5.2e+129)
		tmp = t_1;
	elseif (x <= -2.6e+72)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (x <= -7.2e+15)
		tmp = t_1;
	elseif (x <= -8e-73)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (x <= 4.8e-124)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((z / (y_m / x)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -1.05e+207)
		tmp = t_0;
	elseif (x <= -5.2e+129)
		tmp = t_1;
	elseif (x <= -2.6e+72)
		tmp = abs((z * (x / y_m)));
	elseif (x <= -7.2e+15)
		tmp = t_1;
	elseif (x <= -8e-73)
		tmp = abs((x * (z / y_m)));
	elseif (x <= 4.8e-124)
		tmp = abs((4.0 / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.05e+207], t$95$0, If[LessEqual[x, -5.2e+129], t$95$1, If[LessEqual[x, -2.6e+72], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -7.2e+15], t$95$1, If[LessEqual[x, -8e-73], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.8e-124], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.05e207 or 4.79999999999999985e-124 < x

   1. Initial program 91.0%

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

   2. Taylor expanded in z around inf 60.8%

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

      1. mul-1-neg 60.8%

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

      2. associate-*l/ 66.5%

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

      3. distribute-rgt-neg-out 66.5%

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

   4. Simplified 66.5%

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

      1. clear-num 66.5%

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

      2. associate-*l/ 66.6%

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

      3. *-un-lft-identity 66.6%

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

      4. add-sqr-sqrt 33.2%

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

      5. sqrt-unprod 53.0%

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

      6. sqr-neg 53.0%

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

      7. sqrt-unprod 33.3%

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

      8. add-sqr-sqrt 66.6%

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

   6. Applied egg-rr 66.6%

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

   if -1.05e207 < x < -5.20000000000000024e129 or -2.59999999999999981e72 < x < -7.2e15

   1. Initial program 88.3%

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

   2. Taylor expanded in z around 0 83.1%

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

      1. associate-*r/ 83.1%

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

      2. metadata-eval 83.1%

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

   4. Simplified 83.1%

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

   5. Taylor expanded in x around inf 83.1%

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

   if -5.20000000000000024e129 < x < -2.59999999999999981e72

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 64.3%

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

      1. mul-1-neg 64.3%

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

      2. associate-*l/ 88.1%

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

      3. distribute-rgt-neg-out 88.1%

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

   4. Simplified 88.1%

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

      1. add-sqr-sqrt 25.4%

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

      2. sqrt-unprod 52.6%

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

      3. sqr-neg 52.6%

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

      4. sqrt-unprod 62.0%

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

      5. add-sqr-sqrt 88.1%

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

      6. expm1-log1p-u 48.4%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y} \cdot z\right)\right)}\right| \]

      7. expm1-udef 24.3%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{y} \cdot z\right)} - 1}\right| \]

      8. associate-*l/ 13.5%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot z}{y}}\right)} - 1\right| \]

   6. Applied egg-rr 13.5%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)} - 1}\right| \]
   7. Step-by-step derivation

      1. expm1-def 37.3%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)\right)}\right| \]

      2. expm1-log1p 64.3%

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

      3. associate-*l/ 88.1%

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

   8. Simplified 88.1%

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

   if -7.2e15 < x < -7.99999999999999998e-73

   1. Initial program 95.0%

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

   2. Taylor expanded in z around inf 68.6%

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

      1. mul-1-neg 68.6%

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

      2. associate-*l/ 64.0%

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

      3. distribute-rgt-neg-out 64.0%

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

   4. Simplified 64.0%

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

      1. add-sqr-sqrt 21.1%

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

      2. sqrt-unprod 40.4%

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

      3. sqr-neg 40.4%

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

      4. sqrt-unprod 42.6%

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

      5. add-sqr-sqrt 64.0%

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

      6. associate-*l/ 68.6%

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

      7. associate-/l* 68.6%

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

   6. Applied egg-rr 68.6%

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

      1. clear-num 68.6%

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

      2. associate-/r/ 68.8%

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

      3. clear-num 68.8%

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

   8. Applied egg-rr 68.8%

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

   if -7.99999999999999998e-73 < x < 4.79999999999999985e-124

   1. Initial program 97.9%

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

   2. Taylor expanded in x around 0 85.2%

      \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+207}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{+129}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{+72}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-73}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 6: 65.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|\frac{z}{\frac{y_m}{x}}\right|\\ t_1 := \left|\frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{+207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{+72}:\\ \;\;\;\;\left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-123}:\\ \;\;\;\;\left|\frac{x \cdot z}{y_m}\right|\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (/ z (/ y_m x)))) (t_1 (fabs (/ x y_m))))
   (if (<= x -1.15e+207)
     t_0
     (if (<= x -1.7e+129)
       t_1
       (if (<= x -8.2e+72)
         (fabs (* z (/ x y_m)))
         (if (<= x -8.2e+15)
           t_1
           (if (<= x -6e-123)
             (fabs (/ (* x z) y_m))
             (if (<= x 5.1e-124) (fabs (/ 4.0 y_m)) t_0))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z / (y_m / x)));
	double t_1 = fabs((x / y_m));
	double tmp;
	if (x <= -1.15e+207) {
		tmp = t_0;
	} else if (x <= -1.7e+129) {
		tmp = t_1;
	} else if (x <= -8.2e+72) {
		tmp = fabs((z * (x / y_m)));
	} else if (x <= -8.2e+15) {
		tmp = t_1;
	} else if (x <= -6e-123) {
		tmp = fabs(((x * z) / y_m));
	} else if (x <= 5.1e-124) {
		tmp = fabs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z / (y_m / x)))
    t_1 = abs((x / y_m))
    if (x <= (-1.15d+207)) then
        tmp = t_0
    else if (x <= (-1.7d+129)) then
        tmp = t_1
    else if (x <= (-8.2d+72)) then
        tmp = abs((z * (x / y_m)))
    else if (x <= (-8.2d+15)) then
        tmp = t_1
    else if (x <= (-6d-123)) then
        tmp = abs(((x * z) / y_m))
    else if (x <= 5.1d-124) then
        tmp = abs((4.0d0 / y_m))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((z / (y_m / x)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (x <= -1.15e+207) {
		tmp = t_0;
	} else if (x <= -1.7e+129) {
		tmp = t_1;
	} else if (x <= -8.2e+72) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (x <= -8.2e+15) {
		tmp = t_1;
	} else if (x <= -6e-123) {
		tmp = Math.abs(((x * z) / y_m));
	} else if (x <= 5.1e-124) {
		tmp = Math.abs((4.0 / y_m));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z / (y_m / x)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if x <= -1.15e+207:
		tmp = t_0
	elif x <= -1.7e+129:
		tmp = t_1
	elif x <= -8.2e+72:
		tmp = math.fabs((z * (x / y_m)))
	elif x <= -8.2e+15:
		tmp = t_1
	elif x <= -6e-123:
		tmp = math.fabs(((x * z) / y_m))
	elif x <= 5.1e-124:
		tmp = math.fabs((4.0 / y_m))
	else:
		tmp = t_0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z / Float64(y_m / x)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -1.15e+207)
		tmp = t_0;
	elseif (x <= -1.7e+129)
		tmp = t_1;
	elseif (x <= -8.2e+72)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (x <= -8.2e+15)
		tmp = t_1;
	elseif (x <= -6e-123)
		tmp = abs(Float64(Float64(x * z) / y_m));
	elseif (x <= 5.1e-124)
		tmp = abs(Float64(4.0 / y_m));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((z / (y_m / x)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -1.15e+207)
		tmp = t_0;
	elseif (x <= -1.7e+129)
		tmp = t_1;
	elseif (x <= -8.2e+72)
		tmp = abs((z * (x / y_m)));
	elseif (x <= -8.2e+15)
		tmp = t_1;
	elseif (x <= -6e-123)
		tmp = abs(((x * z) / y_m));
	elseif (x <= 5.1e-124)
		tmp = abs((4.0 / y_m));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.15e+207], t$95$0, If[LessEqual[x, -1.7e+129], t$95$1, If[LessEqual[x, -8.2e+72], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -8.2e+15], t$95$1, If[LessEqual[x, -6e-123], N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5.1e-124], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.14999999999999997e207 or 5.1000000000000001e-124 < x

   1. Initial program 91.0%

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

   2. Taylor expanded in z around inf 60.8%

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

      1. mul-1-neg 60.8%

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

      2. associate-*l/ 66.5%

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

      3. distribute-rgt-neg-out 66.5%

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

   4. Simplified 66.5%

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

      1. clear-num 66.5%

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

      2. associate-*l/ 66.6%

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

      3. *-un-lft-identity 66.6%

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

      4. add-sqr-sqrt 33.2%

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

      5. sqrt-unprod 53.0%

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

      6. sqr-neg 53.0%

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

      7. sqrt-unprod 33.3%

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

      8. add-sqr-sqrt 66.6%

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

   6. Applied egg-rr 66.6%

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

   if -1.14999999999999997e207 < x < -1.70000000000000009e129 or -8.19999999999999926e72 < x < -8.2e15

   1. Initial program 88.3%

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

   2. Taylor expanded in z around 0 83.1%

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

      1. associate-*r/ 83.1%

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

      2. metadata-eval 83.1%

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

   4. Simplified 83.1%

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

   5. Taylor expanded in x around inf 83.1%

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

   if -1.70000000000000009e129 < x < -8.19999999999999926e72

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 64.3%

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

      1. mul-1-neg 64.3%

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

      2. associate-*l/ 88.1%

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

      3. distribute-rgt-neg-out 88.1%

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

   4. Simplified 88.1%

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

      1. add-sqr-sqrt 25.4%

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

      2. sqrt-unprod 52.6%

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

      3. sqr-neg 52.6%

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

      4. sqrt-unprod 62.0%

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

      5. add-sqr-sqrt 88.1%

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

      6. expm1-log1p-u 48.4%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y} \cdot z\right)\right)}\right| \]

      7. expm1-udef 24.3%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{y} \cdot z\right)} - 1}\right| \]

      8. associate-*l/ 13.5%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot z}{y}}\right)} - 1\right| \]

   6. Applied egg-rr 13.5%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)} - 1}\right| \]
   7. Step-by-step derivation

      1. expm1-def 37.3%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot z}{y}\right)\right)}\right| \]

      2. expm1-log1p 64.3%

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

      3. associate-*l/ 88.1%

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

   8. Simplified 88.1%

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

   if -8.2e15 < x < -5.99999999999999968e-123

   1. Initial program 96.8%

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

   2. Taylor expanded in z around inf 64.5%

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

      1. mul-1-neg 64.5%

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

      2. associate-*l/ 61.5%

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

      3. distribute-rgt-neg-out 61.5%

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

   4. Simplified 61.5%

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

      1. add-sqr-sqrt 23.8%

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

      2. sqrt-unprod 37.1%

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

      3. sqr-neg 37.1%

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

      4. sqrt-unprod 37.4%

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

      5. add-sqr-sqrt 61.5%

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

      6. associate-*l/ 64.5%

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

   6. Applied egg-rr 64.5%

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

   if -5.99999999999999968e-123 < x < 5.1000000000000001e-124

   1. Initial program 97.6%

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

   2. Taylor expanded in x around 0 90.3%

      \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+207}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{+129}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{+72}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{+15}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-123}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-124}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 7: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 2.65 \cdot 10^{-45}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y_m} - \frac{x}{\frac{y_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2.65e-45)
   (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
   (fabs (- (/ (+ 4.0 x) 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 <= 2.65e-45) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = fabs((((4.0 + x) / 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 <= 2.65d-45) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    else
        tmp = abs((((4.0d0 + x) / 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 <= 2.65e-45) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((4.0 + x) / 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 <= 2.65e-45:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((4.0 + x) / 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 <= 2.65e-45)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(4.0 + x) / 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 <= 2.65e-45)
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	else
		tmp = abs((((4.0 + x) / 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, 2.65e-45], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(4.0 + x), $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 < 2.6499999999999999e-45

   1. Initial program 91.9%

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

   2. Taylor expanded in y around 0 96.3%

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

   if 2.6499999999999999e-45 < y

   1. Initial program 98.6%

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

      1. associate-/r/ 99.8%

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

   3. Applied egg-rr 99.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-45}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 8: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+38}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y_m} + \frac{-1}{y_m}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -1.2e+38)
   (fabs (* x (+ (/ z y_m) (/ -1.0 y_m))))
   (fabs (/ (- (+ 4.0 x) (* x z)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.2e+38) {
		tmp = fabs((x * ((z / y_m) + (-1.0 / y_m))));
	} else {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.2d+38)) then
        tmp = abs((x * ((z / y_m) + ((-1.0d0) / y_m))))
    else
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -1.2e+38) {
		tmp = Math.abs((x * ((z / y_m) + (-1.0 / y_m))));
	} else {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -1.2e+38:
		tmp = math.fabs((x * ((z / y_m) + (-1.0 / y_m))))
	else:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -1.2e+38)
		tmp = abs(Float64(x * Float64(Float64(z / y_m) + Float64(-1.0 / y_m))));
	else
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -1.2e+38)
		tmp = abs((x * ((z / y_m) + (-1.0 / y_m))));
	else
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -1.2e+38], N[Abs[N[(x * N[(N[(z / y$95$m), $MachinePrecision] + N[(-1.0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.20000000000000009e38

   1. Initial program 89.4%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in x around inf 99.8%

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

   if -1.20000000000000009e38 < x

   1. Initial program 95.0%

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

   2. Taylor expanded in y around 0 98.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+38}:\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y} + \frac{-1}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 9: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+37}:\\ \;\;\;\;\left|\frac{z}{\frac{y_m}{x}}\right|\\ \mathbf{elif}\;z \leq 2500000000000:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -2.2e+37)
   (fabs (/ z (/ y_m x)))
   (if (<= z 2500000000000.0)
     (fabs (/ (- -4.0 x) y_m))
     (fabs (* x (/ z y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -2.2e+37) {
		tmp = fabs((z / (y_m / x)));
	} else if (z <= 2500000000000.0) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((x * (z / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.2d+37)) then
        tmp = abs((z / (y_m / x)))
    else if (z <= 2500000000000.0d0) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((x * (z / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -2.2e+37) {
		tmp = Math.abs((z / (y_m / x)));
	} else if (z <= 2500000000000.0) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((x * (z / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -2.2e+37:
		tmp = math.fabs((z / (y_m / x)))
	elif z <= 2500000000000.0:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((x * (z / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -2.2e+37)
		tmp = abs(Float64(z / Float64(y_m / x)));
	elseif (z <= 2500000000000.0)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(x * Float64(z / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -2.2e+37)
		tmp = abs((z / (y_m / x)));
	elseif (z <= 2500000000000.0)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((x * (z / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -2.2e+37], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2500000000000.0], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2000000000000001e37

   1. Initial program 98.1%

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

   2. Taylor expanded in z around inf 69.8%

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

      1. mul-1-neg 69.8%

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

      2. associate-*l/ 78.2%

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

      3. distribute-rgt-neg-out 78.2%

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

   4. Simplified 78.2%

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

      1. clear-num 78.1%

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

      2. associate-*l/ 78.3%

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

      3. *-un-lft-identity 78.3%

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

      4. add-sqr-sqrt 78.0%

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

      5. sqrt-unprod 54.5%

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

      6. sqr-neg 54.5%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 78.3%

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

   6. Applied egg-rr 78.3%

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

   if -2.2000000000000001e37 < z < 2.5e12

   1. Initial program 96.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 96.8%

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

      1. associate-*r/ 96.8%

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

      2. distribute-lft-in 96.8%

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

      3. metadata-eval 96.8%

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

      4. neg-mul-1 96.8%

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

      5. sub-neg 96.8%

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

   6. Simplified 96.8%

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

   if 2.5e12 < z

   1. Initial program 84.7%

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

   2. Taylor expanded in z around inf 79.3%

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

      1. mul-1-neg 79.3%

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

      2. associate-*l/ 80.6%

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

      3. distribute-rgt-neg-out 80.6%

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

   4. Simplified 80.6%

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot \left(-z\right)}\right| \]
   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 57.7%

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

      3. sqr-neg 57.7%

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

      4. sqrt-unprod 80.4%

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

      5. add-sqr-sqrt 80.6%

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

      6. associate-*l/ 79.3%

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

      7. associate-/l* 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. clear-num 83.2%

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

      2. associate-/r/ 83.2%

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

      3. clear-num 83.2%

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

   8. Applied egg-rr 83.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+37}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;z \leq 2500000000000:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 10: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -10.5) (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 <= -10.5) || !(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 <= (-10.5d0)) .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 <= -10.5) || !(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 <= -10.5) 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 <= -10.5) || !(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 <= -10.5) || ~((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, -10.5], 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 < -10.5 or 4 < x

   1. Initial program 89.5%

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

   2. Taylor expanded in z around 0 58.8%

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

      1. associate-*r/ 58.8%

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

      2. metadata-eval 58.8%

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

   4. Simplified 58.8%

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

   5. Taylor expanded in x around inf 58.2%

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

   if -10.5 < x < 4

   1. Initial program 97.8%

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

   2. Taylor expanded in x around 0 70.3%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 11: 40.6% 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 93.7%

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

2. Taylor expanded in x around 0 38.3%

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

3. Final simplification 38.3%

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

Reproduce

herbie shell --seed 2023332 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))