fabs fraction 1

Percentage Accurate: 92.2% → 99.9%

Time: 7.9s

Alternatives: 10

Speedup: 1.0×

• 21.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.2% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.5e-27)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (fma x (/ z y) (/ (- -4.0 x) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.5e-27) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(fma(x, (z / y), ((-4.0 - x) / y)));
	}
	return tmp;
}

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.5e-27)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	end
	return tmp
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 1.5e-27], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5000000000000001e-27

   1. Initial program 88.0%

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

      1. associate-*l/ 88.8%

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

      2. sub-div 96.7%

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

   3. Applied egg-rr 96.7%

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

   if 1.5000000000000001e-27 < y

   1. Initial program 98.2%

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

   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 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 2: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \frac{x + 4}{y} - z \cdot \frac{x}{y}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ (+ x 4.0) y) (* z (/ x y)))))
   (if (<= t_0 4e+306) (fabs t_0) (fabs (/ (- (+ x 4.0) (* x z)) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = ((x + 4.0) / y) - (z * (x / y));
	double tmp;
	if (t_0 <= 4e+306) {
		tmp = fabs(t_0);
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + 4.0d0) / y) - (z * (x / y))
    if (t_0 <= 4d+306) then
        tmp = abs(t_0)
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = ((x + 4.0) / y) - (z * (x / y));
	double tmp;
	if (t_0 <= 4e+306) {
		tmp = Math.abs(t_0);
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = ((x + 4.0) / y) - (z * (x / y))
	tmp = 0
	if t_0 <= 4e+306:
		tmp = math.fabs(t_0)
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = Float64(Float64(Float64(x + 4.0) / y) - Float64(z * Float64(x / y)))
	tmp = 0.0
	if (t_0 <= 4e+306)
		tmp = abs(t_0);
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = ((x + 4.0) / y) - (z * (x / y));
	tmp = 0.0;
	if (t_0 <= 4e+306)
		tmp = abs(t_0);
	else
		tmp = abs((((x + 4.0) - (x * z)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e+306], N[Abs[t$95$0], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)) < 4.00000000000000007e306

   1. Initial program 99.7%

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

   if 4.00000000000000007e306 < (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))

   1. Initial program 48.8%

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

      1. associate-*l/ 65.1%

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

      2. sub-div 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - z \cdot \frac{x}{y} \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 3: 64.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+245}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{+172}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-78} \lor \neg \left(x \leq 0.00033\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* x (/ z y)))))
   (if (<= x -2.5e+245)
     t_0
     (if (<= x -2.1e+172)
       (fabs (/ x y))
       (if (or (<= x -8e-78) (not (<= x 0.00033))) t_0 (fabs (/ 4.0 y)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((x * (z / y)));
	double tmp;
	if (x <= -2.5e+245) {
		tmp = t_0;
	} else if (x <= -2.1e+172) {
		tmp = fabs((x / y));
	} else if ((x <= -8e-78) || !(x <= 0.00033)) {
		tmp = t_0;
	} else {
		tmp = fabs((4.0 / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x * (z / y)))
    if (x <= (-2.5d+245)) then
        tmp = t_0
    else if (x <= (-2.1d+172)) then
        tmp = abs((x / y))
    else if ((x <= (-8d-78)) .or. (.not. (x <= 0.00033d0))) then
        tmp = t_0
    else
        tmp = abs((4.0d0 / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double t_0 = Math.abs((x * (z / y)));
	double tmp;
	if (x <= -2.5e+245) {
		tmp = t_0;
	} else if (x <= -2.1e+172) {
		tmp = Math.abs((x / y));
	} else if ((x <= -8e-78) || !(x <= 0.00033)) {
		tmp = t_0;
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((x * (z / y)))
	tmp = 0
	if x <= -2.5e+245:
		tmp = t_0
	elif x <= -2.1e+172:
		tmp = math.fabs((x / y))
	elif (x <= -8e-78) or not (x <= 0.00033):
		tmp = t_0
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(x * Float64(z / y)))
	tmp = 0.0
	if (x <= -2.5e+245)
		tmp = t_0;
	elseif (x <= -2.1e+172)
		tmp = abs(Float64(x / y));
	elseif ((x <= -8e-78) || !(x <= 0.00033))
		tmp = t_0;
	else
		tmp = abs(Float64(4.0 / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((x * (z / y)));
	tmp = 0.0;
	if (x <= -2.5e+245)
		tmp = t_0;
	elseif (x <= -2.1e+172)
		tmp = abs((x / y));
	elseif ((x <= -8e-78) || ~((x <= 0.00033)))
		tmp = t_0;
	else
		tmp = abs((4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.5e+245], t$95$0, If[LessEqual[x, -2.1e+172], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, -8e-78], N[Not[LessEqual[x, 0.00033]], $MachinePrecision]], t$95$0, N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000017e245 or -2.1000000000000001e172 < x < -7.99999999999999999e-78 or 3.3e-4 < x

   1. Initial program 86.4%

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

   2. Taylor expanded in z around inf 57.2%

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

      1. mul-1-neg 57.2%

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

      2. associate-*r/ 69.7%

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

      3. distribute-rgt-neg-out 69.7%

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

      4. distribute-neg-frac 69.7%

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

   4. Simplified 69.7%

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

      1. add-sqr-sqrt 35.6%

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

      2. sqrt-unprod 54.3%

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

      3. sqr-neg 54.3%

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

      4. sqrt-unprod 33.9%

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

      5. add-sqr-sqrt 69.7%

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

      6. clear-num 69.6%

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

      7. expm1-log1p-u 35.3%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\frac{y}{z}}\right)\right)}\right| \]

      8. div-inv 35.3%

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

      9. expm1-udef 29.4%

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

      10. associate-/r/ 31.8%

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

      11. associate-/r/ 29.4%

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

      12. div-inv 29.4%

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

      13. clear-num 29.4%

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

   6. Applied egg-rr 29.4%

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

      1. expm1-def 35.4%

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

      2. expm1-log1p 69.7%

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

   8. Simplified 69.7%

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

   if -2.50000000000000017e245 < x < -2.1000000000000001e172

   1. Initial program 71.4%

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

   2. Taylor expanded in z around 0 82.1%

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

      1. associate-*r/ 82.1%

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

      2. metadata-eval 82.1%

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

   4. Simplified 82.1%

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

   5. Taylor expanded in x around inf 82.1%

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

   if -7.99999999999999999e-78 < x < 3.3e-4

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 86.1%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+245}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{+172}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-78} \lor \neg \left(x \leq 0.00033\right):\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 4: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.07\right):\\ \;\;\;\;\left|\frac{z}{y} \cdot \left(\frac{4}{z} - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 0.07)))
   (fabs (* (/ z y) (- (/ 4.0 z) x)))
   (fabs (/ (- -4.0 x) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.07)) {
		tmp = fabs(((z / y) * ((4.0 / z) - x)));
	} else {
		tmp = fabs(((-4.0 - x) / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 0.07d0))) then
        tmp = abs(((z / y) * ((4.0d0 / z) - x)))
    else
        tmp = abs((((-4.0d0) - x) / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.07)) {
		tmp = Math.abs(((z / y) * ((4.0 / z) - x)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.07):
		tmp = math.fabs(((z / y) * ((4.0 / z) - x)))
	else:
		tmp = math.fabs(((-4.0 - x) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.07))
		tmp = abs(Float64(Float64(z / y) * Float64(Float64(4.0 / z) - x)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 0.07)))
		tmp = abs(((z / y) * ((4.0 / z) - x)));
	else
		tmp = abs(((-4.0 - x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.07]], $MachinePrecision]], N[Abs[N[(N[(z / y), $MachinePrecision] * N[(N[(4.0 / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.070000000000000007 < z

   1. Initial program 92.2%

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

      1. associate-/r/ 81.7%

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

      2. frac-sub 56.4%

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

      3. associate-/r* 68.4%

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

      4. *-commutative 68.4%

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

   3. Applied egg-rr 68.4%

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

   4. Taylor expanded in x around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. associate-*l/ 68.1%

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

   6. Simplified 68.1%

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

   7. Taylor expanded in y around 0 87.5%

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

      1. associate-/l* 94.3%

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

      2. associate-/r/ 88.3%

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

      3. associate-*r/ 88.3%

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

      4. metadata-eval 88.3%

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

   9. Simplified 88.3%

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

   if -1 < z < 0.070000000000000007

   1. Initial program 90.0%

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

   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 99.8%

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

      1. associate-*r/ 99.8%

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

      2. distribute-lft-in 99.8%

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

      3. metadata-eval 99.8%

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

      4. neg-mul-1 99.8%

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

      5. sub-neg 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.07\right):\\ \;\;\;\;\left|\frac{z}{y} \cdot \left(\frac{4}{z} - x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-10)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ (+ x 4.0) y) (/ x (/ y z))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-10) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((((x + 4.0) / y) - (x / (y / z))));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2d-10) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs((((x + 4.0d0) / y) - (x / (y / z))))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-10) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((((x + 4.0) / y) - (x / (y / z))));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 2e-10:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((((x + 4.0) / y) - (x / (y / z))))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-10)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(x / Float64(y / z))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-10)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((((x + 4.0) / y) - (x / (y / z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 2e-10], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000007e-10

   1. Initial program 87.7%

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

      1. associate-*l/ 88.4%

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

      2. sub-div 96.8%

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

   3. Applied egg-rr 96.8%

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

   if 2.00000000000000007e-10 < y

   1. Initial program 99.4%

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

   3. Simplified 99.9%

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

4. Final simplification 97.7%

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

Alternative 6: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+224}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 2.2e+224) (fabs (/ (- (+ x 4.0) (* x z)) y)) (fabs (* x (/ z y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.2e+224) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((x * (z / y)));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 2.2d+224) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs((x * (z / y)))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 2.2e+224) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((x * (z / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if x <= 2.2e+224:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((x * (z / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (x <= 2.2e+224)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(x * Float64(z / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 2.2e+224)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((x * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 2.2e+224], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2e224

   1. Initial program 91.3%

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

      1. associate-*l/ 90.9%

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

      2. sub-div 96.3%

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

   3. Applied egg-rr 96.3%

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

   if 2.2e224 < x

   1. Initial program 88.8%

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

   2. Taylor expanded in z around inf 47.8%

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

      1. mul-1-neg 47.8%

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

      2. associate-*r/ 89.5%

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

      3. distribute-rgt-neg-out 89.5%

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

      4. distribute-neg-frac 89.5%

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

   4. Simplified 89.5%

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

      1. add-sqr-sqrt 44.7%

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

      2. sqrt-unprod 78.8%

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

      3. sqr-neg 78.8%

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

      4. sqrt-unprod 44.5%

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

      5. add-sqr-sqrt 89.5%

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

      6. clear-num 89.3%

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

      7. expm1-log1p-u 42.8%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\frac{y}{z}}\right)\right)}\right| \]

      8. div-inv 42.8%

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

      9. expm1-udef 42.8%

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

      10. associate-/r/ 42.9%

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

      11. associate-/r/ 42.8%

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

      12. div-inv 42.8%

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

      13. clear-num 42.9%

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

   6. Applied egg-rr 42.9%

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

      1. expm1-def 42.9%

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

      2. expm1-log1p 89.5%

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

   8. Simplified 89.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+224}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 7: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-78} \lor \neg \left(x \leq 7.2 \cdot 10^{-6}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9e-78) (not (<= x 7.2e-6)))
   (fabs (* z (/ x y)))
   (fabs (/ 4.0 y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9e-78) || !(x <= 7.2e-6)) {
		tmp = fabs((z * (x / y)));
	} else {
		tmp = fabs((4.0 / y));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-9d-78)) .or. (.not. (x <= 7.2d-6))) then
        tmp = abs((z * (x / y)))
    else
        tmp = abs((4.0d0 / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9e-78) || !(x <= 7.2e-6)) {
		tmp = Math.abs((z * (x / y)));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -9e-78) or not (x <= 7.2e-6):
		tmp = math.fabs((z * (x / y)))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -9e-78) || !(x <= 7.2e-6))
		tmp = abs(Float64(z * Float64(x / y)));
	else
		tmp = abs(Float64(4.0 / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -9e-78) || ~((x <= 7.2e-6)))
		tmp = abs((z * (x / y)));
	else
		tmp = abs((4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[x, -9e-78], N[Not[LessEqual[x, 7.2e-6]], $MachinePrecision]], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9e-78 or 7.19999999999999967e-6 < x

   1. Initial program 84.1%

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

   2. Taylor expanded in z around inf 56.8%

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

      1. mul-1-neg 56.8%

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

      2. associate-*r/ 68.1%

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

      3. distribute-rgt-neg-out 68.1%

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

      4. distribute-neg-frac 68.1%

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

   4. Simplified 68.1%

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

      1. add-sqr-sqrt 35.4%

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

      2. sqrt-unprod 54.9%

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

      3. sqr-neg 54.9%

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

      4. sqrt-unprod 32.5%

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

      5. add-sqr-sqrt 68.1%

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

      6. clear-num 68.0%

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

      7. expm1-log1p-u 32.3%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\frac{y}{z}}\right)\right)}\right| \]

      8. div-inv 32.3%

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

      9. expm1-udef 27.2%

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

      10. associate-/r/ 29.9%

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

      11. associate-/r/ 27.2%

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

      12. div-inv 27.2%

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

      13. clear-num 27.2%

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

   6. Applied egg-rr 27.2%

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

      1. expm1-def 32.3%

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

      2. expm1-log1p 68.1%

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

      3. associate-*r/ 56.8%

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

      4. associate-*l/ 74.2%

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

      5. *-commutative 74.2%

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

   8. Simplified 74.2%

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

   if -9e-78 < x < 7.19999999999999967e-6

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 86.1%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-78} \lor \neg \left(x \leq 7.2 \cdot 10^{-6}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 8: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1550000000:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+56}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z -1550000000.0)
   (fabs (* z (/ x y)))
   (if (<= z 3.9e+56) (fabs (/ (- -4.0 x) y)) (fabs (/ z (/ y x))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1550000000.0) {
		tmp = fabs((z * (x / y)));
	} else if (z <= 3.9e+56) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((z / (y / x)));
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1550000000.0d0)) then
        tmp = abs((z * (x / y)))
    else if (z <= 3.9d+56) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((z / (y / x)))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1550000000.0) {
		tmp = Math.abs((z * (x / y)));
	} else if (z <= 3.9e+56) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((z / (y / x)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if z <= -1550000000.0:
		tmp = math.fabs((z * (x / y)))
	elif z <= 3.9e+56:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((z / (y / x)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (z <= -1550000000.0)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (z <= 3.9e+56)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(z / Float64(y / x)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1550000000.0)
		tmp = abs((z * (x / y)));
	elseif (z <= 3.9e+56)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((z / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[z, -1550000000.0], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.9e+56], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.55e9

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 71.4%

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

      1. mul-1-neg 71.4%

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

      2. associate-*r/ 78.0%

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

      3. distribute-rgt-neg-out 78.0%

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

      4. distribute-neg-frac 78.0%

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

   4. Simplified 78.0%

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

      1. add-sqr-sqrt 77.8%

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

      2. sqrt-unprod 61.2%

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

      3. sqr-neg 61.2%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 78.0%

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

      6. clear-num 77.8%

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

      7. expm1-log1p-u 44.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{\frac{y}{z}}\right)\right)}\right| \]

      8. div-inv 44.7%

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

      9. expm1-udef 36.6%

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

      10. associate-/r/ 41.4%

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

      11. associate-/r/ 36.6%

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

      12. div-inv 36.6%

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

      13. clear-num 36.6%

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

   6. Applied egg-rr 36.6%

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

      1. expm1-def 44.6%

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

      2. expm1-log1p 78.0%

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

      3. associate-*r/ 71.4%

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

      4. associate-*l/ 84.7%

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

      5. *-commutative 84.7%

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

   8. Simplified 84.7%

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

   if -1.55e9 < z < 3.89999999999999994e56

   1. Initial program 90.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in z around 0 97.1%

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

      1. associate-*r/ 97.1%

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

      2. distribute-lft-in 97.1%

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

      3. metadata-eval 97.1%

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

      4. neg-mul-1 97.1%

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

      5. sub-neg 97.1%

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

   6. Simplified 97.1%

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

   if 3.89999999999999994e56 < z

   1. Initial program 82.2%

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

   2. Taylor expanded in z around inf 66.5%

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

      1. mul-1-neg 66.5%

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

      2. associate-*r/ 74.0%

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

      3. distribute-rgt-neg-out 74.0%

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

      4. distribute-neg-frac 74.0%

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

   4. Simplified 74.0%

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

      1. associate-*r/ 66.5%

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

      2. add-cube-cbrt 65.9%

         \[\leadsto \left|\frac{x \cdot \left(-z\right)}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\right| \]

      3. times-frac 74.6%

         \[\leadsto \left|\color{blue}{\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{-z}{\sqrt[3]{y}}}\right| \]

      4. add-sqr-sqrt 0.0%

         \[\leadsto \left|\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}{\sqrt[3]{y}}\right| \]

      5. sqrt-unprod 51.6%

         \[\leadsto \left|\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}{\sqrt[3]{y}}\right| \]

      6. sqr-neg 51.6%

         \[\leadsto \left|\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt{\color{blue}{z \cdot z}}}{\sqrt[3]{y}}\right| \]

      7. sqrt-unprod 74.5%

         \[\leadsto \left|\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}{\sqrt[3]{y}}\right| \]

      8. add-sqr-sqrt 74.6%

         \[\leadsto \left|\frac{x}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\color{blue}{z}}{\sqrt[3]{y}}\right| \]

      9. times-frac 65.9%

         \[\leadsto \left|\color{blue}{\frac{x \cdot z}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\right| \]

      10. *-commutative 65.9%

          \[\leadsto \left|\frac{\color{blue}{z \cdot x}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right| \]

      11. add-cube-cbrt 66.5%

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

      12. associate-/l* 77.0%

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

   6. Applied egg-rr 77.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1550000000:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+56}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 9: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.52 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.52) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.52d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y))
    else
        tmp = abs((4.0d0 / y))
    end if
    code = tmp
end function

Java

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

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -1.52) or not (x <= 4.0):
		tmp = math.fabs((x / y))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.52) || !(x <= 4.0))
		tmp = abs(Float64(x / y));
	else
		tmp = abs(Float64(4.0 / y));
	end
	return tmp
end

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[x, -1.52], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.52 or 4 < x

   1. Initial program 82.1%

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

   2. Taylor expanded in z around 0 57.3%

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

      1. associate-*r/ 57.3%

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

      2. metadata-eval 57.3%

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

   4. Simplified 57.3%

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

   5. Taylor expanded in x around inf 56.5%

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

   if -1.52 < x < 4

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 80.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 10: 39.2% accurate, 1.1× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z) :precision binary64 (fabs (/ 4.0 y)))

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = abs((4.0d0 / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 91.2%

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

2. Taylor expanded in x around 0 44.2%

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

3. Final simplification 44.2%

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

Reproduce

herbie shell --seed 2023314 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))