fabs fraction 1

Percentage Accurate: 92.4% → 99.9%

Time: 7.9s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.4% 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

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\left|\frac{4 + \left(1 - z\right) \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-8)
   (fabs (/ (+ 4.0 (* (- 1.0 z) x)) 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 <= 2e-8) {
		tmp = fabs(((4.0 + ((1.0 - z) * x)) / 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 <= 2e-8)
		tmp = abs(Float64(Float64(4.0 + Float64(Float64(1.0 - z) * x)) / 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, 2e-8], N[Abs[N[(N[(4.0 + N[(N[(1.0 - z), $MachinePrecision] * x), $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 < 2e-8

   1. Initial program 90.2%

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

      1. fabs-neg 90.2%

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

      2. sub-neg 90.2%

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

      3. distribute-neg-in 90.2%

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

      4. sub-neg 90.2%

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

      5. distribute-neg-frac 90.2%

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

      6. associate-*l/ 92.7%

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

      7. distribute-neg-frac 92.7%

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

      8. neg-mul-1 92.7%

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

      9. associate-*l/ 92.6%

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

      10. neg-mul-1 92.6%

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

      11. associate-*l/ 92.5%

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

      12. distribute-lft-out-- 97.8%

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

      13. fabs-mul 97.8%

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

      14. fabs-sub 97.8%

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

      15. fabs-mul 97.8%

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

      16. associate-*l/ 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in x around 0 98.0%

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

   if 2e-8 < y

   1. Initial program 90.0%

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

      1. fabs-sub 90.0%

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

      2. associate-*l/ 95.6%

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

      3. *-commutative 95.6%

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

      4. associate-*l/ 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-neg 99.7%

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

      7. distribute-neg-frac 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\left|\frac{4 + \left(1 - z\right) \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]

Alternative 2: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+33}:\\ \;\;\;\;\frac{\left|x\right|}{\left|\frac{y}{z}\right|}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+14}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.25e+33)
   (/ (fabs x) (fabs (/ y z)))
   (if (<= z 1.3e+14) (fabs (/ (- -4.0 x) y)) (fabs (/ (* z x) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.25e+33) {
		tmp = fabs(x) / fabs((y / z));
	} else if (z <= 1.3e+14) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs(((z * 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 <= (-1.25d+33)) then
        tmp = abs(x) / abs((y / z))
    else if (z <= 1.3d+14) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs(((z * 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 <= -1.25e+33) {
		tmp = Math.abs(x) / Math.abs((y / z));
	} else if (z <= 1.3e+14) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs(((z * x) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if z <= -1.25e+33:
		tmp = math.fabs(x) / math.fabs((y / z))
	elif z <= 1.3e+14:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs(((z * x) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.25e+33)
		tmp = Float64(abs(x) / abs(Float64(y / z)));
	elseif (z <= 1.3e+14)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(Float64(z * x) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.25e+33)
		tmp = abs(x) / abs((y / z));
	elseif (z <= 1.3e+14)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs(((z * x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[z, -1.25e+33], N[(N[Abs[x], $MachinePrecision] / N[Abs[N[(y / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.3e+14], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.24999999999999993e33

   1. Initial program 92.9%

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

   2. Taylor expanded in z around inf 77.0%

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

      1. *-commutative 77.0%

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

      2. metadata-eval 77.0%

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

      3. times-frac 77.0%

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

      4. *-lft-identity 77.0%

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

      5. neg-mul-1 77.0%

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

      6. associate-/l* 83.3%

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

      7. distribute-frac-neg 83.3%

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

   4. Simplified 83.3%

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

   if -1.24999999999999993e33 < z < 1.3e14

   1. Initial program 95.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 95.9%

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

      1. associate-*r/ 95.9%

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

      2. distribute-lft-in 95.9%

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

      3. metadata-eval 95.9%

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

      4. neg-mul-1 95.9%

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

      5. sub-neg 95.9%

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

   6. Simplified 95.9%

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

   if 1.3e14 < z

   1. Initial program 75.7%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in z around inf 77.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+33}:\\ \;\;\;\;\frac{\left|x\right|}{\left|\frac{y}{z}\right|}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+14}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot x}{y}\right|\\ \end{array} \]

Alternative 3: 67.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|x \cdot \frac{z}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -3.3 \cdot 10^{+225}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.96 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.55:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-54}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+182}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* x (/ z y)))) (t_1 (fabs (/ x y))))
   (if (<= x -3.3e+225)
     t_0
     (if (<= x -1.1e+99)
       t_1
       (if (<= x -1.96e+59)
         t_0
         (if (<= x -1.55)
           t_1
           (if (<= x 7e-54)
             (fabs (/ 4.0 y))
             (if (<= x 2.3e+182) t_0 t_1))))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((x * (z / y)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -3.3e+225) {
		tmp = t_0;
	} else if (x <= -1.1e+99) {
		tmp = t_1;
	} else if (x <= -1.96e+59) {
		tmp = t_0;
	} else if (x <= -1.55) {
		tmp = t_1;
	} else if (x <= 7e-54) {
		tmp = fabs((4.0 / y));
	} else if (x <= 2.3e+182) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = abs((x * (z / y)))
    t_1 = abs((x / y))
    if (x <= (-3.3d+225)) then
        tmp = t_0
    else if (x <= (-1.1d+99)) then
        tmp = t_1
    else if (x <= (-1.96d+59)) then
        tmp = t_0
    else if (x <= (-1.55d0)) then
        tmp = t_1
    else if (x <= 7d-54) then
        tmp = abs((4.0d0 / y))
    else if (x <= 2.3d+182) then
        tmp = t_0
    else
        tmp = t_1
    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 t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -3.3e+225) {
		tmp = t_0;
	} else if (x <= -1.1e+99) {
		tmp = t_1;
	} else if (x <= -1.96e+59) {
		tmp = t_0;
	} else if (x <= -1.55) {
		tmp = t_1;
	} else if (x <= 7e-54) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 2.3e+182) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((x * (z / y)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -3.3e+225:
		tmp = t_0
	elif x <= -1.1e+99:
		tmp = t_1
	elif x <= -1.96e+59:
		tmp = t_0
	elif x <= -1.55:
		tmp = t_1
	elif x <= 7e-54:
		tmp = math.fabs((4.0 / y))
	elif x <= 2.3e+182:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(x * Float64(z / y)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -3.3e+225)
		tmp = t_0;
	elseif (x <= -1.1e+99)
		tmp = t_1;
	elseif (x <= -1.96e+59)
		tmp = t_0;
	elseif (x <= -1.55)
		tmp = t_1;
	elseif (x <= 7e-54)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 2.3e+182)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((x * (z / y)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -3.3e+225)
		tmp = t_0;
	elseif (x <= -1.1e+99)
		tmp = t_1;
	elseif (x <= -1.96e+59)
		tmp = t_0;
	elseif (x <= -1.55)
		tmp = t_1;
	elseif (x <= 7e-54)
		tmp = abs((4.0 / y));
	elseif (x <= 2.3e+182)
		tmp = t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.3e+225], t$95$0, If[LessEqual[x, -1.1e+99], t$95$1, If[LessEqual[x, -1.96e+59], t$95$0, If[LessEqual[x, -1.55], t$95$1, If[LessEqual[x, 7e-54], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.3e+182], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.3e225 or -1.09999999999999989e99 < x < -1.96000000000000007e59 or 6.99999999999999964e-54 < x < 2.3e182

   1. Initial program 92.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in z around inf 64.6%

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

      1. associate-*l/ 68.0%

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

      2. *-commutative 68.0%

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

   6. Simplified 68.0%

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

   if -3.3e225 < x < -1.09999999999999989e99 or -1.96000000000000007e59 < x < -1.55000000000000004 or 2.3e182 < x

   1. Initial program 87.7%

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

   2. Taylor expanded in z around 0 74.7%

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

      1. associate-*r/ 74.7%

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

      2. metadata-eval 74.7%

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

   4. Simplified 74.7%

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

   5. Taylor expanded in x around inf 72.8%

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

   if -1.55000000000000004 < x < 6.99999999999999964e-54

   1. Initial program 90.1%

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

   2. Taylor expanded in x around 0 72.7%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+225}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{+99}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.96 \cdot 10^{+59}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq -1.55:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-54}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+182}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+177}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-54}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))) (t_1 (fabs (/ x y))))
   (if (<= x -1.25e+177)
     t_0
     (if (<= x -2.7e+98)
       t_1
       (if (<= x -4.4e+57)
         t_0
         (if (<= x -1.5) t_1 (if (<= x 8e-54) (fabs (/ 4.0 y)) t_0)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -1.25e+177) {
		tmp = t_0;
	} else if (x <= -2.7e+98) {
		tmp = t_1;
	} else if (x <= -4.4e+57) {
		tmp = t_0;
	} else if (x <= -1.5) {
		tmp = t_1;
	} else if (x <= 8e-54) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = t_0;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y)))
    t_1 = abs((x / y))
    if (x <= (-1.25d+177)) then
        tmp = t_0
    else if (x <= (-2.7d+98)) then
        tmp = t_1
    else if (x <= (-4.4d+57)) then
        tmp = t_0
    else if (x <= (-1.5d0)) then
        tmp = t_1
    else if (x <= 8d-54) then
        tmp = abs((4.0d0 / y))
    else
        tmp = t_0
    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((z * (x / y)));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -1.25e+177) {
		tmp = t_0;
	} else if (x <= -2.7e+98) {
		tmp = t_1;
	} else if (x <= -4.4e+57) {
		tmp = t_0;
	} else if (x <= -1.5) {
		tmp = t_1;
	} else if (x <= 8e-54) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -1.25e+177:
		tmp = t_0
	elif x <= -2.7e+98:
		tmp = t_1
	elif x <= -4.4e+57:
		tmp = t_0
	elif x <= -1.5:
		tmp = t_1
	elif x <= 8e-54:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = t_0
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -1.25e+177)
		tmp = t_0;
	elseif (x <= -2.7e+98)
		tmp = t_1;
	elseif (x <= -4.4e+57)
		tmp = t_0;
	elseif (x <= -1.5)
		tmp = t_1;
	elseif (x <= 8e-54)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -1.25e+177)
		tmp = t_0;
	elseif (x <= -2.7e+98)
		tmp = t_1;
	elseif (x <= -4.4e+57)
		tmp = t_0;
	elseif (x <= -1.5)
		tmp = t_1;
	elseif (x <= 8e-54)
		tmp = abs((4.0 / y));
	else
		tmp = t_0;
	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[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.25e+177], t$95$0, If[LessEqual[x, -2.7e+98], t$95$1, If[LessEqual[x, -4.4e+57], t$95$0, If[LessEqual[x, -1.5], t$95$1, If[LessEqual[x, 8e-54], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2500000000000001e177 or -2.7e98 < x < -4.4000000000000001e57 or 8.0000000000000002e-54 < x

   1. Initial program 88.7%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in z around inf 58.8%

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

      1. *-commutative 58.8%

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

      2. associate-*l/ 69.3%

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

      3. *-commutative 69.3%

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

   6. Simplified 69.3%

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

   if -1.2500000000000001e177 < x < -2.7e98 or -4.4000000000000001e57 < x < -1.5

   1. Initial program 96.4%

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

   2. Taylor expanded in z around 0 78.3%

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

      1. associate-*r/ 78.3%

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

      2. metadata-eval 78.3%

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

   4. Simplified 78.3%

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

   5. Taylor expanded in x around inf 74.0%

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

   if -1.5 < x < 8.0000000000000002e-54

   1. Initial program 90.1%

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

   2. Taylor expanded in x around 0 72.7%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+177}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{+57}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.5:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-54}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 5: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{+178}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.18 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{+57}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq -1.55:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-54}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))) (t_1 (fabs (/ x y))))
   (if (<= x -6.6e+178)
     t_0
     (if (<= x -1.18e+98)
       t_1
       (if (<= x -1.85e+57)
         (fabs (/ z (/ y x)))
         (if (<= x -1.55) t_1 (if (<= x 8.5e-54) (fabs (/ 4.0 y)) t_0)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double t_1 = fabs((x / y));
	double tmp;
	if (x <= -6.6e+178) {
		tmp = t_0;
	} else if (x <= -1.18e+98) {
		tmp = t_1;
	} else if (x <= -1.85e+57) {
		tmp = fabs((z / (y / x)));
	} else if (x <= -1.55) {
		tmp = t_1;
	} else if (x <= 8.5e-54) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = t_0;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y)))
    t_1 = abs((x / y))
    if (x <= (-6.6d+178)) then
        tmp = t_0
    else if (x <= (-1.18d+98)) then
        tmp = t_1
    else if (x <= (-1.85d+57)) then
        tmp = abs((z / (y / x)))
    else if (x <= (-1.55d0)) then
        tmp = t_1
    else if (x <= 8.5d-54) then
        tmp = abs((4.0d0 / y))
    else
        tmp = t_0
    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((z * (x / y)));
	double t_1 = Math.abs((x / y));
	double tmp;
	if (x <= -6.6e+178) {
		tmp = t_0;
	} else if (x <= -1.18e+98) {
		tmp = t_1;
	} else if (x <= -1.85e+57) {
		tmp = Math.abs((z / (y / x)));
	} else if (x <= -1.55) {
		tmp = t_1;
	} else if (x <= 8.5e-54) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	t_1 = math.fabs((x / y))
	tmp = 0
	if x <= -6.6e+178:
		tmp = t_0
	elif x <= -1.18e+98:
		tmp = t_1
	elif x <= -1.85e+57:
		tmp = math.fabs((z / (y / x)))
	elif x <= -1.55:
		tmp = t_1
	elif x <= 8.5e-54:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = t_0
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	t_1 = abs(Float64(x / y))
	tmp = 0.0
	if (x <= -6.6e+178)
		tmp = t_0;
	elseif (x <= -1.18e+98)
		tmp = t_1;
	elseif (x <= -1.85e+57)
		tmp = abs(Float64(z / Float64(y / x)));
	elseif (x <= -1.55)
		tmp = t_1;
	elseif (x <= 8.5e-54)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	t_1 = abs((x / y));
	tmp = 0.0;
	if (x <= -6.6e+178)
		tmp = t_0;
	elseif (x <= -1.18e+98)
		tmp = t_1;
	elseif (x <= -1.85e+57)
		tmp = abs((z / (y / x)));
	elseif (x <= -1.55)
		tmp = t_1;
	elseif (x <= 8.5e-54)
		tmp = abs((4.0 / y));
	else
		tmp = t_0;
	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[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -6.6e+178], t$95$0, If[LessEqual[x, -1.18e+98], t$95$1, If[LessEqual[x, -1.85e+57], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -1.55], t$95$1, If[LessEqual[x, 8.5e-54], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.5999999999999996e178 or 8.5e-54 < x

   1. Initial program 88.4%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in z around inf 56.8%

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

      1. *-commutative 56.8%

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

      2. associate-*l/ 68.5%

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

      3. *-commutative 68.5%

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

   6. Simplified 68.5%

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

   if -6.5999999999999996e178 < x < -1.18000000000000002e98 or -1.85000000000000003e57 < x < -1.55000000000000004

   1. Initial program 96.4%

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

   2. Taylor expanded in z around 0 78.3%

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

      1. associate-*r/ 78.3%

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

      2. metadata-eval 78.3%

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

   4. Simplified 78.3%

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

   5. Taylor expanded in x around inf 74.0%

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

   if -1.18000000000000002e98 < x < -1.85000000000000003e57

   1. Initial program 91.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in z around inf 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-*l/ 76.3%

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

      3. *-commutative 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in z around 0 76.4%

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

      1. associate-/l* 76.4%

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

   9. Simplified 76.4%

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

   if -1.55000000000000004 < x < 8.5e-54

   1. Initial program 90.1%

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

   2. Taylor expanded in x around 0 72.7%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+178}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.18 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{+57}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq -1.55:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-54}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 6: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 4.2\right):\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 - z \cdot x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.6) (not (<= x 4.2)))
   (fabs (/ (- 1.0 z) (/ y x)))
   (fabs (/ (- 4.0 (* z x)) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.6) || !(x <= 4.2)) {
		tmp = fabs(((1.0 - z) / (y / x)));
	} else {
		tmp = fabs(((4.0 - (z * 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 ((x <= (-1.6d0)) .or. (.not. (x <= 4.2d0))) then
        tmp = abs(((1.0d0 - z) / (y / x)))
    else
        tmp = abs(((4.0d0 - (z * 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 ((x <= -1.6) || !(x <= 4.2)) {
		tmp = Math.abs(((1.0 - z) / (y / x)));
	} else {
		tmp = Math.abs(((4.0 - (z * x)) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -1.6) or not (x <= 4.2):
		tmp = math.fabs(((1.0 - z) / (y / x)))
	else:
		tmp = math.fabs(((4.0 - (z * x)) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.6) || !(x <= 4.2))
		tmp = abs(Float64(Float64(1.0 - z) / Float64(y / x)));
	else
		tmp = abs(Float64(Float64(4.0 - Float64(z * x)) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.6) || ~((x <= 4.2)))
		tmp = abs(((1.0 - z) / (y / x)));
	else
		tmp = abs(((4.0 - (z * x)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[Or[LessEqual[x, -1.6], N[Not[LessEqual[x, 4.2]], $MachinePrecision]], N[Abs[N[(N[(1.0 - z), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(4.0 - N[(z * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001 or 4.20000000000000018 < x

   1. Initial program 89.4%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in x around inf 93.3%

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

      1. associate-/l* 98.1%

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

      2. sub-neg 98.1%

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

      3. metadata-eval 98.1%

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

   6. Simplified 98.1%

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

   if -1.6000000000000001 < x < 4.20000000000000018

   1. Initial program 90.9%

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

      1. fabs-neg 90.9%

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

      2. sub-neg 90.9%

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

      3. distribute-neg-in 90.9%

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

      4. sub-neg 90.9%

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

      5. distribute-neg-frac 90.9%

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

      6. associate-*l/ 99.9%

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

      7. distribute-neg-frac 99.9%

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

      8. neg-mul-1 99.9%

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

      9. associate-*l/ 99.9%

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

      10. neg-mul-1 99.9%

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

      11. associate-*l/ 99.8%

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

      12. distribute-lft-out-- 99.8%

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

      13. fabs-mul 99.8%

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

      14. fabs-sub 99.8%

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

      15. fabs-mul 99.8%

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

      16. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

   5. Taylor expanded in z around inf 98.5%

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

      1. neg-mul-1 98.5%

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

      2. distribute-lft-neg-in 98.5%

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

      3. *-commutative 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 4.2\right):\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 - z \cdot x}{y}\right|\\ \end{array} \]

Alternative 7: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+31}:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + \left(1 - z\right) \cdot x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x -1.45e+31)
   (fabs (/ (- 1.0 z) (/ y x)))
   (fabs (/ (+ 4.0 (* (- 1.0 z) x)) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e+31) {
		tmp = fabs(((1.0 - z) / (y / x)));
	} else {
		tmp = fabs(((4.0 + ((1.0 - z) * 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 (x <= (-1.45d+31)) then
        tmp = abs(((1.0d0 - z) / (y / x)))
    else
        tmp = abs(((4.0d0 + ((1.0d0 - z) * 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 (x <= -1.45e+31) {
		tmp = Math.abs(((1.0 - z) / (y / x)));
	} else {
		tmp = Math.abs(((4.0 + ((1.0 - z) * x)) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if x <= -1.45e+31:
		tmp = math.fabs(((1.0 - z) / (y / x)))
	else:
		tmp = math.fabs(((4.0 + ((1.0 - z) * x)) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (x <= -1.45e+31)
		tmp = abs(Float64(Float64(1.0 - z) / Float64(y / x)));
	else
		tmp = abs(Float64(Float64(4.0 + Float64(Float64(1.0 - z) * x)) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.45e+31)
		tmp = abs(((1.0 - z) / (y / x)));
	else
		tmp = abs(((4.0 + ((1.0 - z) * x)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, -1.45e+31], N[Abs[N[(N[(1.0 - z), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(4.0 + N[(N[(1.0 - z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45e31

   1. Initial program 87.8%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in x around inf 92.9%

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

      1. associate-/l* 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

   if -1.45e31 < x

   1. Initial program 91.0%

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

      1. fabs-neg 91.0%

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

      2. sub-neg 91.0%

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

      3. distribute-neg-in 91.0%

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

      4. sub-neg 91.0%

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

      5. distribute-neg-frac 91.0%

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

      6. associate-*l/ 97.3%

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

      7. distribute-neg-frac 97.3%

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

      8. neg-mul-1 97.3%

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

      9. associate-*l/ 97.2%

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

      10. neg-mul-1 97.2%

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

      11. associate-*l/ 97.2%

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

      12. distribute-lft-out-- 98.8%

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

      13. fabs-mul 98.8%

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

      14. fabs-sub 98.8%

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

      15. fabs-mul 98.8%

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

      16. associate-*l/ 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in x around 0 98.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+31}:\\ \;\;\;\;\left|\frac{1 - z}{\frac{y}{x}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + \left(1 - z\right) \cdot x}{y}\right|\\ \end{array} \]

Alternative 8: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{+32}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+14}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= z -6.1e+32)
   (fabs (* x (/ z y)))
   (if (<= z 2.6e+14) (fabs (/ (- -4.0 x) y)) (fabs (/ (* z x) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.1e+32) {
		tmp = fabs((x * (z / y)));
	} else if (z <= 2.6e+14) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs(((z * 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 <= (-6.1d+32)) then
        tmp = abs((x * (z / y)))
    else if (z <= 2.6d+14) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs(((z * 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 <= -6.1e+32) {
		tmp = Math.abs((x * (z / y)));
	} else if (z <= 2.6e+14) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs(((z * x) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if z <= -6.1e+32:
		tmp = math.fabs((x * (z / y)))
	elif z <= 2.6e+14:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs(((z * x) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (z <= -6.1e+32)
		tmp = abs(Float64(x * Float64(z / y)));
	elseif (z <= 2.6e+14)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(Float64(z * x) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6.1e+32)
		tmp = abs((x * (z / y)));
	elseif (z <= 2.6e+14)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs(((z * x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[z, -6.1e+32], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 2.6e+14], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.10000000000000027e32

   1. Initial program 92.9%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in z around inf 77.0%

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

      1. associate-*l/ 83.3%

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

      2. *-commutative 83.3%

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

   6. Simplified 83.3%

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

   if -6.10000000000000027e32 < z < 2.6e14

   1. Initial program 95.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 95.9%

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

      1. associate-*r/ 95.9%

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

      2. distribute-lft-in 95.9%

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

      3. metadata-eval 95.9%

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

      4. neg-mul-1 95.9%

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

      5. sub-neg 95.9%

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

   6. Simplified 95.9%

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

   if 2.6e14 < z

   1. Initial program 75.7%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in z around inf 77.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.1 \cdot 10^{+32}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+14}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot x}{y}\right|\\ \end{array} \]

Alternative 9: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.5) (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 <= -1.5) || !(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 <= (-1.5d0)) .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 <= -1.5) || !(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 <= -1.5) 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 <= -1.5) || !(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 <= -1.5) || ~((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, -1.5], 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 < -1.5 or 4 < x

   1. Initial program 89.4%

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

   2. Taylor expanded in z around 0 60.5%

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

      1. associate-*r/ 60.5%

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

      2. metadata-eval 60.5%

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

   4. Simplified 60.5%

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

   5. Taylor expanded in x around inf 58.8%

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

   if -1.5 < x < 4

   1. Initial program 90.9%

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

   2. Taylor expanded in x around 0 70.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 10: 40.9% 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 90.1%

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

2. Taylor expanded in x around 0 36.0%

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

3. Final simplification 36.0%

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

Reproduce

herbie shell --seed 2023252 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))