fabs fraction 1

Percentage Accurate: 91.9% → 99.8%

Time: 7.6s

Alternatives: 10

Speedup: 1.0×

• 21.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y_m}, \frac{-4 - x}{y_m}\right)\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2e-37)
   (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 <= 2e-37) {
		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 <= 2e-37)
		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, 2e-37], 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 < 2.00000000000000013e-37

   1. Initial program 86.2%

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

   2. Taylor expanded in y around 0 97.4%

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

   if 2.00000000000000013e-37 < y

   1. Initial program 99.8%

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

   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 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 2: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -8 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-120}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+110} \lor \neg \left(x \leq 3.2 \cdot 10^{+229}\right) \land x \leq 1.1 \cdot 10^{+284}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))))
   (if (<= x -8e-16)
     t_0
     (if (<= x 3.4e-120)
       (fabs (/ 4.0 y_m))
       (if (or (<= x 9e+110) (and (not (<= x 3.2e+229)) (<= x 1.1e+284)))
         t_0
         (fabs (/ x 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 tmp;
	if (x <= -8e-16) {
		tmp = t_0;
	} else if (x <= 3.4e-120) {
		tmp = fabs((4.0 / y_m));
	} else if ((x <= 9e+110) || (!(x <= 3.2e+229) && (x <= 1.1e+284))) {
		tmp = t_0;
	} else {
		tmp = fabs((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) :: t_0
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    if (x <= (-8d-16)) then
        tmp = t_0
    else if (x <= 3.4d-120) then
        tmp = abs((4.0d0 / y_m))
    else if ((x <= 9d+110) .or. (.not. (x <= 3.2d+229)) .and. (x <= 1.1d+284)) then
        tmp = t_0
    else
        tmp = abs((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 t_0 = Math.abs((z * (x / y_m)));
	double tmp;
	if (x <= -8e-16) {
		tmp = t_0;
	} else if (x <= 3.4e-120) {
		tmp = Math.abs((4.0 / y_m));
	} else if ((x <= 9e+110) || (!(x <= 3.2e+229) && (x <= 1.1e+284))) {
		tmp = t_0;
	} else {
		tmp = Math.abs((x / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z * (x / y_m)))
	tmp = 0
	if x <= -8e-16:
		tmp = t_0
	elif x <= 3.4e-120:
		tmp = math.fabs((4.0 / y_m))
	elif (x <= 9e+110) or (not (x <= 3.2e+229) and (x <= 1.1e+284)):
		tmp = t_0
	else:
		tmp = math.fabs((x / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z * Float64(x / y_m)))
	tmp = 0.0
	if (x <= -8e-16)
		tmp = t_0;
	elseif (x <= 3.4e-120)
		tmp = abs(Float64(4.0 / y_m));
	elseif ((x <= 9e+110) || (!(x <= 3.2e+229) && (x <= 1.1e+284)))
		tmp = t_0;
	else
		tmp = abs(Float64(x / 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)));
	tmp = 0.0;
	if (x <= -8e-16)
		tmp = t_0;
	elseif (x <= 3.4e-120)
		tmp = abs((4.0 / y_m));
	elseif ((x <= 9e+110) || (~((x <= 3.2e+229)) && (x <= 1.1e+284)))
		tmp = t_0;
	else
		tmp = abs((x / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -8e-16], t$95$0, If[LessEqual[x, 3.4e-120], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 9e+110], And[N[Not[LessEqual[x, 3.2e+229]], $MachinePrecision], LessEqual[x, 1.1e+284]]], t$95$0, N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.9999999999999998e-16 or 3.4000000000000001e-120 < x < 9.0000000000000005e110 or 3.1999999999999998e229 < x < 1.09999999999999997e284

   1. Initial program 86.8%

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

   2. Taylor expanded in z around inf 52.9%

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

      1. mul-1-neg 52.9%

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

      2. associate-*l/ 64.4%

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

      3. distribute-rgt-neg-out 64.4%

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

   4. Simplified 64.4%

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

      1. expm1-log1p-u 33.0%

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

      2. expm1-udef 27.8%

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

      3. *-commutative 27.8%

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

      4. add-sqr-sqrt 14.5%

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

      5. sqrt-unprod 26.9%

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

      6. sqr-neg 26.9%

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

      7. sqrt-unprod 16.8%

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

      8. add-sqr-sqrt 31.2%

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

      9. *-commutative 31.2%

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

      10. associate-*l/ 26.5%

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

      11. associate-/l* 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. expm1-def 34.7%

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

      2. expm1-log1p 60.1%

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

      3. associate-/r/ 64.4%

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

   8. Simplified 64.4%

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

   if -7.9999999999999998e-16 < x < 3.4000000000000001e-120

   1. Initial program 97.6%

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

   2. Taylor expanded in x around 0 85.5%

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

   if 9.0000000000000005e110 < x < 3.1999999999999998e229 or 1.09999999999999997e284 < x

   1. Initial program 78.0%

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

   2. Taylor expanded in z around 0 83.7%

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

      1. associate-*r/ 83.7%

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

      2. metadata-eval 83.7%

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

   4. Simplified 83.7%

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

   5. Taylor expanded in x around inf 83.7%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-16}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-120}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+110} \lor \neg \left(x \leq 3.2 \cdot 10^{+229}\right) \land x \leq 1.1 \cdot 10^{+284}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-120}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+104}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{z}}\right|\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+227} \lor \neg \left(x \leq 1.16 \cdot 10^{+284}\right):\\ \;\;\;\;\left|\frac{x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))))
   (if (<= x -9.8e-14)
     t_0
     (if (<= x 3.4e-120)
       (fabs (/ 4.0 y_m))
       (if (<= x 1.5e+104)
         (fabs (/ x (/ y_m z)))
         (if (or (<= x 5.2e+227) (not (<= x 1.16e+284)))
           (fabs (/ x 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 tmp;
	if (x <= -9.8e-14) {
		tmp = t_0;
	} else if (x <= 3.4e-120) {
		tmp = fabs((4.0 / y_m));
	} else if (x <= 1.5e+104) {
		tmp = fabs((x / (y_m / z)));
	} else if ((x <= 5.2e+227) || !(x <= 1.16e+284)) {
		tmp = fabs((x / 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) :: tmp
    t_0 = abs((z * (x / y_m)))
    if (x <= (-9.8d-14)) then
        tmp = t_0
    else if (x <= 3.4d-120) then
        tmp = abs((4.0d0 / y_m))
    else if (x <= 1.5d+104) then
        tmp = abs((x / (y_m / z)))
    else if ((x <= 5.2d+227) .or. (.not. (x <= 1.16d+284))) then
        tmp = abs((x / 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 tmp;
	if (x <= -9.8e-14) {
		tmp = t_0;
	} else if (x <= 3.4e-120) {
		tmp = Math.abs((4.0 / y_m));
	} else if (x <= 1.5e+104) {
		tmp = Math.abs((x / (y_m / z)));
	} else if ((x <= 5.2e+227) || !(x <= 1.16e+284)) {
		tmp = Math.abs((x / 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)))
	tmp = 0
	if x <= -9.8e-14:
		tmp = t_0
	elif x <= 3.4e-120:
		tmp = math.fabs((4.0 / y_m))
	elif x <= 1.5e+104:
		tmp = math.fabs((x / (y_m / z)))
	elif (x <= 5.2e+227) or not (x <= 1.16e+284):
		tmp = math.fabs((x / 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)))
	tmp = 0.0
	if (x <= -9.8e-14)
		tmp = t_0;
	elseif (x <= 3.4e-120)
		tmp = abs(Float64(4.0 / y_m));
	elseif (x <= 1.5e+104)
		tmp = abs(Float64(x / Float64(y_m / z)));
	elseif ((x <= 5.2e+227) || !(x <= 1.16e+284))
		tmp = abs(Float64(x / 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)));
	tmp = 0.0;
	if (x <= -9.8e-14)
		tmp = t_0;
	elseif (x <= 3.4e-120)
		tmp = abs((4.0 / y_m));
	elseif (x <= 1.5e+104)
		tmp = abs((x / (y_m / z)));
	elseif ((x <= 5.2e+227) || ~((x <= 1.16e+284)))
		tmp = abs((x / 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]}, If[LessEqual[x, -9.8e-14], t$95$0, If[LessEqual[x, 3.4e-120], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.5e+104], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 5.2e+227], N[Not[LessEqual[x, 1.16e+284]], $MachinePrecision]], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.79999999999999989e-14 or 5.19999999999999964e227 < x < 1.1599999999999999e284

   1. Initial program 83.0%

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

   2. Taylor expanded in z around inf 52.1%

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

      1. mul-1-neg 52.1%

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

      2. associate-*l/ 69.3%

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

      3. distribute-rgt-neg-out 69.3%

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

   4. Simplified 69.3%

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

      1. expm1-log1p-u 31.9%

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

      2. expm1-udef 29.0%

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

      3. *-commutative 29.0%

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

      4. add-sqr-sqrt 15.6%

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

      5. sqrt-unprod 29.3%

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

      6. sqr-neg 29.3%

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

      7. sqrt-unprod 18.3%

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

      8. add-sqr-sqrt 36.7%

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

      9. *-commutative 36.7%

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

      10. associate-*l/ 29.8%

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

      11. associate-/l* 33.0%

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

   6. Applied egg-rr 33.0%

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

      1. expm1-def 35.8%

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

      2. expm1-log1p 61.8%

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

      3. associate-/r/ 69.3%

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

   8. Simplified 69.3%

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

   if -9.79999999999999989e-14 < x < 3.4000000000000001e-120

   1. Initial program 97.6%

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

   2. Taylor expanded in x around 0 85.5%

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

   if 3.4000000000000001e-120 < x < 1.49999999999999984e104

   1. Initial program 94.5%

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

   2. Taylor expanded in z around inf 54.6%

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

      1. mul-1-neg 54.6%

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

      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 26.9%

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

      2. sqrt-unprod 39.3%

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

      3. sqr-neg 39.3%

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

      4. sqrt-unprod 27.6%

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

      5. add-sqr-sqrt 54.5%

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

      6. associate-*l/ 54.6%

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

      7. associate-/l* 56.6%

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

   6. Applied egg-rr 56.6%

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

   if 1.49999999999999984e104 < x < 5.19999999999999964e227 or 1.1599999999999999e284 < x

   1. Initial program 78.0%

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

   2. Taylor expanded in z around 0 83.7%

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

      1. associate-*r/ 83.7%

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

      2. metadata-eval 83.7%

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

   4. Simplified 83.7%

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

   5. Taylor expanded in x around inf 83.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-14}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-120}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+104}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+227} \lor \neg \left(x \leq 1.16 \cdot 10^{+284}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+26} \lor \neg \left(x \leq 9.5 \cdot 10^{+51}\right):\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y_m} + \frac{-1}{y_m}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -4.6e+26) (not (<= x 9.5e+51)))
   (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 <= -4.6e+26) || !(x <= 9.5e+51)) {
		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 <= (-4.6d+26)) .or. (.not. (x <= 9.5d+51))) 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 <= -4.6e+26) || !(x <= 9.5e+51)) {
		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 <= -4.6e+26) or not (x <= 9.5e+51):
		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 <= -4.6e+26) || !(x <= 9.5e+51))
		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 <= -4.6e+26) || ~((x <= 9.5e+51)))
		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[Or[LessEqual[x, -4.6e+26], N[Not[LessEqual[x, 9.5e+51]], $MachinePrecision]], 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 < -4.6000000000000001e26 or 9.4999999999999999e51 < x

   1. Initial program 79.7%

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

   3. Simplified 93.8%

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

   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 -4.6000000000000001e26 < x < 9.4999999999999999e51

   1. Initial program 97.5%

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

   2. Taylor expanded in y around 0 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+26} \lor \neg \left(x \leq 9.5 \cdot 10^{+51}\right):\\ \;\;\;\;\left|x \cdot \left(\frac{z}{y} + \frac{-1}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 5: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.5 \cdot 10^{-43}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y_m} - \frac{4 + x}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 1.5e-43)
   (fabs (/ (- (+ 4.0 x) (* x z)) y_m))
   (fabs (- (* z (/ x 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-43) {
		tmp = fabs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = fabs(((z * (x / y_m)) - ((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 (y_m <= 1.5d-43) then
        tmp = abs((((4.0d0 + x) - (x * z)) / y_m))
    else
        tmp = abs(((z * (x / y_m)) - ((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 (y_m <= 1.5e-43) {
		tmp = Math.abs((((4.0 + x) - (x * z)) / y_m));
	} else {
		tmp = Math.abs(((z * (x / y_m)) - ((4.0 + x) / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 1.5e-43:
		tmp = math.fabs((((4.0 + x) - (x * z)) / y_m))
	else:
		tmp = math.fabs(((z * (x / 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-43)
		tmp = abs(Float64(Float64(Float64(4.0 + x) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(z * Float64(x / y_m)) - 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 (y_m <= 1.5e-43)
		tmp = abs((((4.0 + x) - (x * z)) / y_m));
	else
		tmp = abs(((z * (x / y_m)) - ((4.0 + x) / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 1.5e-43], N[Abs[N[(N[(N[(4.0 + x), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(4.0 + x), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000002e-43

   1. Initial program 86.1%

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

   2. Taylor expanded in y around 0 97.4%

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

   if 1.50000000000000002e-43 < y

   1. Initial program 99.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-43}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y} - \frac{4 + x}{y}\right|\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 6.8 \cdot 10^{-37}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y_m} - \frac{x}{\frac{y_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 6.8e-37)
   (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 <= 6.8e-37) {
		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 <= 6.8d-37) 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 <= 6.8e-37) {
		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 <= 6.8e-37:
		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 <= 6.8e-37)
		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 <= 6.8e-37)
		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, 6.8e-37], 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 < 6.80000000000000037e-37

   1. Initial program 86.2%

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

   2. Taylor expanded in y around 0 97.4%

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

   if 6.80000000000000037e-37 < y

   1. Initial program 99.8%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-37}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 7: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+59}:\\ \;\;\;\;\left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+127}:\\ \;\;\;\;\left|\frac{x}{y_m} + \frac{4}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -5.8e+59)
   (fabs (* z (/ x y_m)))
   (if (<= z 3.4e+127)
     (fabs (+ (/ x y_m) (/ 4.0 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 <= -5.8e+59) {
		tmp = fabs((z * (x / y_m)));
	} else if (z <= 3.4e+127) {
		tmp = fabs(((x / y_m) + (4.0 / 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 <= (-5.8d+59)) then
        tmp = abs((z * (x / y_m)))
    else if (z <= 3.4d+127) then
        tmp = abs(((x / y_m) + (4.0d0 / 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 <= -5.8e+59) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (z <= 3.4e+127) {
		tmp = Math.abs(((x / y_m) + (4.0 / 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 <= -5.8e+59:
		tmp = math.fabs((z * (x / y_m)))
	elif z <= 3.4e+127:
		tmp = math.fabs(((x / y_m) + (4.0 / 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 <= -5.8e+59)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (z <= 3.4e+127)
		tmp = abs(Float64(Float64(x / y_m) + Float64(4.0 / 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 <= -5.8e+59)
		tmp = abs((z * (x / y_m)));
	elseif (z <= 3.4e+127)
		tmp = abs(((x / y_m) + (4.0 / 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, -5.8e+59], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.4e+127], N[Abs[N[(N[(x / y$95$m), $MachinePrecision] + N[(4.0 / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.79999999999999981e59

   1. Initial program 97.3%

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

   2. Taylor expanded in z around inf 67.4%

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

      1. mul-1-neg 67.4%

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

      2. associate-*l/ 73.9%

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

      3. distribute-rgt-neg-out 73.9%

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

   4. Simplified 73.9%

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

      1. expm1-log1p-u 41.1%

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

      2. expm1-udef 34.5%

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

      3. *-commutative 34.5%

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

      4. add-sqr-sqrt 34.5%

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

      5. sqrt-unprod 27.1%

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

      6. sqr-neg 27.1%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 32.2%

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

      9. *-commutative 32.2%

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

      10. associate-*l/ 30.7%

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

      11. associate-/l* 29.1%

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

   6. Applied egg-rr 29.1%

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

      1. expm1-def 37.6%

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

      2. expm1-log1p 72.1%

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

      3. associate-/r/ 73.9%

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

   8. Simplified 73.9%

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

   if -5.79999999999999981e59 < z < 3.39999999999999977e127

   1. Initial program 89.0%

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

   2. Taylor expanded in z around 0 94.5%

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

      1. associate-*r/ 94.5%

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

      2. metadata-eval 94.5%

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

   4. Simplified 94.5%

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

   if 3.39999999999999977e127 < z

   1. Initial program 80.6%

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

   2. Taylor expanded in z around inf 76.0%

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

      1. mul-1-neg 76.0%

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

      2. associate-*l/ 86.2%

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

      3. distribute-rgt-neg-out 86.2%

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

   4. Simplified 86.2%

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot \left(-z\right)}\right| \]
   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 57.0%

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

      3. sqr-neg 57.0%

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

      4. sqrt-unprod 86.2%

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

      5. add-sqr-sqrt 86.2%

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

      6. associate-*l/ 76.0%

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

      7. associate-/l* 87.6%

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

   6. Applied egg-rr 87.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+59}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+127}:\\ \;\;\;\;\left|\frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 8: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+59}:\\ \;\;\;\;\left|z \cdot \frac{x}{y_m}\right|\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+127}:\\ \;\;\;\;\left|\frac{-4 - x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y_m}{z}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -7.6e+59)
   (fabs (* z (/ x y_m)))
   (if (<= z 3.6e+127) (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 <= -7.6e+59) {
		tmp = fabs((z * (x / y_m)));
	} else if (z <= 3.6e+127) {
		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 <= (-7.6d+59)) then
        tmp = abs((z * (x / y_m)))
    else if (z <= 3.6d+127) 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 <= -7.6e+59) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (z <= 3.6e+127) {
		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 <= -7.6e+59:
		tmp = math.fabs((z * (x / y_m)))
	elif z <= 3.6e+127:
		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 <= -7.6e+59)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (z <= 3.6e+127)
		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 <= -7.6e+59)
		tmp = abs((z * (x / y_m)));
	elseif (z <= 3.6e+127)
		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, -7.6e+59], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.6e+127], 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 < -7.6000000000000002e59

   1. Initial program 97.3%

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

   2. Taylor expanded in z around inf 67.4%

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

      1. mul-1-neg 67.4%

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

      2. associate-*l/ 73.9%

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

      3. distribute-rgt-neg-out 73.9%

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

   4. Simplified 73.9%

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

      1. expm1-log1p-u 41.1%

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

      2. expm1-udef 34.5%

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

      3. *-commutative 34.5%

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

      4. add-sqr-sqrt 34.5%

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

      5. sqrt-unprod 27.1%

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

      6. sqr-neg 27.1%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 32.2%

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

      9. *-commutative 32.2%

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

      10. associate-*l/ 30.7%

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

      11. associate-/l* 29.1%

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

   6. Applied egg-rr 29.1%

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

      1. expm1-def 37.6%

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

      2. expm1-log1p 72.1%

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

      3. associate-/r/ 73.9%

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

   8. Simplified 73.9%

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

   if -7.6000000000000002e59 < z < 3.59999999999999979e127

   1. Initial program 89.0%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in z around 0 94.5%

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

      1. associate-*r/ 94.5%

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

      2. distribute-lft-in 94.5%

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

      3. metadata-eval 94.5%

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

      4. neg-mul-1 94.5%

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

      5. sub-neg 94.5%

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

   6. Simplified 94.5%

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

   if 3.59999999999999979e127 < z

   1. Initial program 80.6%

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

   2. Taylor expanded in z around inf 76.0%

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

      1. mul-1-neg 76.0%

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

      2. associate-*l/ 86.2%

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

      3. distribute-rgt-neg-out 86.2%

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

   4. Simplified 86.2%

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot \left(-z\right)}\right| \]
   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 57.0%

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

      3. sqr-neg 57.0%

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

      4. sqrt-unprod 86.2%

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

      5. add-sqr-sqrt 86.2%

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

      6. associate-*l/ 76.0%

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

      7. associate-/l* 87.6%

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

   6. Applied egg-rr 87.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+59}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+127}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 9: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -11 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -11.0) (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 <= -11.0) || !(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 <= (-11.0d0)) .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 <= -11.0) || !(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 <= -11.0) 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 <= -11.0) || !(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 <= -11.0) || ~((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, -11.0], 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 < -11 or 4 < x

   1. Initial program 81.8%

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

   2. Taylor expanded in z around 0 70.3%

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

      1. associate-*r/ 70.3%

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

      2. metadata-eval 70.3%

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

   4. Simplified 70.3%

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

   5. Taylor expanded in x around inf 68.5%

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

   if -11 < x < 4

   1. Initial program 97.3%

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

   2. Taylor expanded in x around 0 73.3%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 10: 40.4% 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 89.6%

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

2. Taylor expanded in x around 0 39.6%

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

3. Final simplification 39.6%

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

Reproduce

herbie shell --seed 2023333 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))