fabs fraction 1

Percentage Accurate: 91.7% → 99.9%

Time: 9.5s

Alternatives: 9

Speedup: 1.0×

• 20.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.7% 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, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 16000000:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 16000000.0)
   (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 <= 16000000.0) {
		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 <= 16000000.0d0) 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 <= 16000000.0) {
		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 <= 16000000.0:
		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 <= 16000000.0)
		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 <= 16000000.0)
		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, 16000000.0], 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 < 1.6e7

   1. Initial program 90.2%

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

      1. associate-*l/ 89.8%

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

      2. associate-*r/ 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in x around 0 94.1%

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

      1. sub-neg 94.1%

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

      2. +-commutative 94.1%

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

      3. distribute-lft-in 86.4%

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

      4. associate-+r+ 86.4%

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

      5. distribute-rgt-in 86.4%

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

      6. associate-*l/ 86.5%

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

      7. *-lft-identity 86.5%

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

      8. +-commutative 86.5%

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

      9. distribute-rgt-neg-out 86.5%

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

      10. sub-neg 86.5%

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

      11. associate-*r/ 89.8%

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

      12. div-sub 97.5%

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

   6. Simplified 97.5%

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

   if 1.6e7 < y

   1. Initial program 92.5%

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

      1. associate-*l/ 87.3%

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

      2. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 16000000:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 2: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -2.75 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+120} \lor \neg \left(x \leq 9.2 \cdot 10^{+175}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))))
   (if (<= x -2.75e-90)
     t_0
     (if (<= x 1.9e-9)
       (fabs (/ 4.0 y))
       (if (or (<= x 4.6e+120) (not (<= x 9.2e+175))) t_0 (fabs (/ x y)))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double tmp;
	if (x <= -2.75e-90) {
		tmp = t_0;
	} else if (x <= 1.9e-9) {
		tmp = fabs((4.0 / y));
	} else if ((x <= 4.6e+120) || !(x <= 9.2e+175)) {
		tmp = t_0;
	} else {
		tmp = fabs((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) :: t_0
    real(8) :: tmp
    t_0 = abs((z * (x / y)))
    if (x <= (-2.75d-90)) then
        tmp = t_0
    else if (x <= 1.9d-9) then
        tmp = abs((4.0d0 / y))
    else if ((x <= 4.6d+120) .or. (.not. (x <= 9.2d+175))) then
        tmp = t_0
    else
        tmp = abs((x / 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((z * (x / y)));
	double tmp;
	if (x <= -2.75e-90) {
		tmp = t_0;
	} else if (x <= 1.9e-9) {
		tmp = Math.abs((4.0 / y));
	} else if ((x <= 4.6e+120) || !(x <= 9.2e+175)) {
		tmp = t_0;
	} else {
		tmp = Math.abs((x / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	tmp = 0
	if x <= -2.75e-90:
		tmp = t_0
	elif x <= 1.9e-9:
		tmp = math.fabs((4.0 / y))
	elif (x <= 4.6e+120) or not (x <= 9.2e+175):
		tmp = t_0
	else:
		tmp = math.fabs((x / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	tmp = 0.0
	if (x <= -2.75e-90)
		tmp = t_0;
	elseif (x <= 1.9e-9)
		tmp = abs(Float64(4.0 / y));
	elseif ((x <= 4.6e+120) || !(x <= 9.2e+175))
		tmp = t_0;
	else
		tmp = abs(Float64(x / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	tmp = 0.0;
	if (x <= -2.75e-90)
		tmp = t_0;
	elseif (x <= 1.9e-9)
		tmp = abs((4.0 / y));
	elseif ((x <= 4.6e+120) || ~((x <= 9.2e+175)))
		tmp = t_0;
	else
		tmp = abs((x / 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[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.75e-90], t$95$0, If[LessEqual[x, 1.9e-9], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 4.6e+120], N[Not[LessEqual[x, 9.2e+175]], $MachinePrecision]], t$95$0, N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.75000000000000015e-90 or 1.90000000000000006e-9 < x < 4.59999999999999985e120 or 9.1999999999999998e175 < x

   1. Initial program 87.0%

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

      1. associate-*l/ 81.5%

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

      2. associate-*r/ 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in z around inf 56.9%

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

      1. associate-*r/ 56.9%

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

      2. neg-mul-1 56.9%

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

      3. distribute-lft-neg-in 56.9%

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

      4. associate-*l/ 69.4%

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

      5. *-commutative 69.4%

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

   6. Simplified 69.4%

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

      1. add-sqr-sqrt 35.4%

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

      2. sqrt-unprod 63.4%

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

      3. distribute-frac-neg 63.4%

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

      4. distribute-frac-neg 63.4%

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

      5. sqr-neg 63.4%

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

      6. sqrt-unprod 33.9%

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

      7. add-sqr-sqrt 69.4%

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

      8. clear-num 69.2%

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

      9. div-inv 67.1%

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

   8. Applied egg-rr 67.1%

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

      1. associate-/r/ 65.7%

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

      2. *-commutative 65.7%

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

      3. add-sqr-sqrt 26.4%

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

      4. sqrt-unprod 61.0%

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

      5. sqr-neg 61.0%

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

      6. sqrt-unprod 39.1%

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

      7. add-sqr-sqrt 65.7%

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

      8. associate-*r/ 56.9%

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

      9. associate-*l/ 69.4%

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

      10. add-sqr-sqrt 40.7%

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

      11. sqrt-unprod 60.6%

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

      12. sqr-neg 60.6%

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

      13. sqrt-unprod 28.4%

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

      14. add-sqr-sqrt 69.4%

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

   10. Applied egg-rr 69.4%

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

   if -2.75000000000000015e-90 < x < 1.90000000000000006e-9

   1. Initial program 97.3%

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

      1. associate-*l/ 99.9%

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

      2. associate-*r/ 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in x around 0 80.7%

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

   if 4.59999999999999985e120 < x < 9.1999999999999998e175

   1. Initial program 75.0%

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

      1. associate-*l/ 83.3%

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

      2. associate-*r/ 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in x around inf 83.3%

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

   5. Taylor expanded in z around 0 88.6%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-90}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+120} \lor \neg \left(x \leq 9.2 \cdot 10^{+175}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]

Alternative 3: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -2.75 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+120}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+176}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))))
   (if (<= x -2.75e-90)
     t_0
     (if (<= x 4e-8)
       (fabs (/ 4.0 y))
       (if (<= x 4.4e+120)
         (fabs (/ z (/ y x)))
         (if (<= x 1.15e+176) (fabs (/ x y)) t_0))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double tmp;
	if (x <= -2.75e-90) {
		tmp = t_0;
	} else if (x <= 4e-8) {
		tmp = fabs((4.0 / y));
	} else if (x <= 4.4e+120) {
		tmp = fabs((z / (y / x)));
	} else if (x <= 1.15e+176) {
		tmp = fabs((x / 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) :: tmp
    t_0 = abs((z * (x / y)))
    if (x <= (-2.75d-90)) then
        tmp = t_0
    else if (x <= 4d-8) then
        tmp = abs((4.0d0 / y))
    else if (x <= 4.4d+120) then
        tmp = abs((z / (y / x)))
    else if (x <= 1.15d+176) then
        tmp = abs((x / 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 tmp;
	if (x <= -2.75e-90) {
		tmp = t_0;
	} else if (x <= 4e-8) {
		tmp = Math.abs((4.0 / y));
	} else if (x <= 4.4e+120) {
		tmp = Math.abs((z / (y / x)));
	} else if (x <= 1.15e+176) {
		tmp = Math.abs((x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	tmp = 0
	if x <= -2.75e-90:
		tmp = t_0
	elif x <= 4e-8:
		tmp = math.fabs((4.0 / y))
	elif x <= 4.4e+120:
		tmp = math.fabs((z / (y / x)))
	elif x <= 1.15e+176:
		tmp = math.fabs((x / y))
	else:
		tmp = t_0
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	tmp = 0.0
	if (x <= -2.75e-90)
		tmp = t_0;
	elseif (x <= 4e-8)
		tmp = abs(Float64(4.0 / y));
	elseif (x <= 4.4e+120)
		tmp = abs(Float64(z / Float64(y / x)));
	elseif (x <= 1.15e+176)
		tmp = abs(Float64(x / 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)));
	tmp = 0.0;
	if (x <= -2.75e-90)
		tmp = t_0;
	elseif (x <= 4e-8)
		tmp = abs((4.0 / y));
	elseif (x <= 4.4e+120)
		tmp = abs((z / (y / x)));
	elseif (x <= 1.15e+176)
		tmp = abs((x / 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]}, If[LessEqual[x, -2.75e-90], t$95$0, If[LessEqual[x, 4e-8], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.4e+120], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.15e+176], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.75000000000000015e-90 or 1.14999999999999998e176 < x

   1. Initial program 84.4%

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

      1. associate-*l/ 80.2%

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

      2. associate-*r/ 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in z around inf 56.9%

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

      1. associate-*r/ 56.9%

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

      2. neg-mul-1 56.9%

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

      3. distribute-lft-neg-in 56.9%

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

      4. associate-*l/ 68.7%

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

      5. *-commutative 68.7%

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

   6. Simplified 68.7%

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

      1. add-sqr-sqrt 35.0%

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

      2. sqrt-unprod 62.5%

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

      3. distribute-frac-neg 62.5%

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

      4. distribute-frac-neg 62.5%

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

      5. sqr-neg 62.5%

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

      6. sqrt-unprod 33.6%

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

      7. add-sqr-sqrt 68.7%

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

      8. clear-num 68.4%

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

      9. div-inv 66.0%

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

   8. Applied egg-rr 66.0%

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

      1. associate-/r/ 65.1%

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

      2. *-commutative 65.1%

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

      3. add-sqr-sqrt 17.9%

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

      4. sqrt-unprod 59.4%

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

      5. sqr-neg 59.4%

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

      6. sqrt-unprod 47.0%

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

      7. add-sqr-sqrt 65.1%

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

      8. associate-*r/ 56.9%

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

      9. associate-*l/ 68.7%

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

      10. add-sqr-sqrt 49.0%

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

      11. sqrt-unprod 58.1%

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

      12. sqr-neg 58.1%

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

      13. sqrt-unprod 19.6%

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

      14. add-sqr-sqrt 68.7%

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

   10. Applied egg-rr 68.7%

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

   if -2.75000000000000015e-90 < x < 4.0000000000000001e-8

   1. Initial program 97.3%

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

      1. associate-*l/ 99.9%

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

      2. associate-*r/ 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in x around 0 80.7%

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

   if 4.0000000000000001e-8 < x < 4.4000000000000003e120

   1. Initial program 99.6%

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

      1. associate-*l/ 87.6%

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

      2. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 56.4%

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

      1. associate-*r/ 56.4%

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

      2. neg-mul-1 56.4%

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

      3. distribute-lft-neg-in 56.4%

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

      4. associate-*l/ 72.7%

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

      5. *-commutative 72.7%

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

   6. Simplified 72.7%

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

      1. add-sqr-sqrt 37.6%

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

      2. sqrt-unprod 68.2%

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

      3. distribute-frac-neg 68.2%

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

      4. distribute-frac-neg 68.2%

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

      5. sqr-neg 68.2%

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

      6. sqrt-unprod 35.2%

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

      7. add-sqr-sqrt 72.7%

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

      8. clear-num 72.7%

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

      9. div-inv 72.9%

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

   8. Applied egg-rr 72.9%

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

   if 4.4000000000000003e120 < x < 1.14999999999999998e176

   1. Initial program 75.0%

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

      1. associate-*l/ 83.3%

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

      2. associate-*r/ 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in x around inf 83.3%

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

   5. Taylor expanded in z around 0 88.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-90}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+120}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+176}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-6)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ (+ x 4.0) y) (* x (/ z y))))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-6) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((((x + 4.0) / y) - (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 (y <= 2d-6) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs((((x + 4.0d0) / y) - (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 (y <= 2e-6) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((((x + 4.0) / y) - (x * (z / y))));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if y <= 2e-6:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((((x + 4.0) / y) - (x * (z / y))))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-6)
		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(z / y))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-6)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((((x + 4.0) / y) - (x * (z / y))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[y, 2e-6], 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[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999991e-6

   1. Initial program 90.0%

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

      1. associate-*l/ 89.6%

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

      2. associate-*r/ 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in x around 0 93.9%

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

      1. sub-neg 93.9%

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

      2. +-commutative 93.9%

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

      3. distribute-lft-in 86.1%

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

      4. associate-+r+ 86.1%

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

      5. distribute-rgt-in 86.1%

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

      6. associate-*l/ 86.2%

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

      7. *-lft-identity 86.2%

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

      8. +-commutative 86.2%

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

      9. distribute-rgt-neg-out 86.2%

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

      10. sub-neg 86.2%

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

      11. associate-*r/ 89.6%

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

      12. div-sub 97.4%

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

   6. Simplified 97.4%

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

   if 1.99999999999999991e-6 < y

   1. Initial program 93.0%

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

      1. associate-*l/ 88.1%

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

      2. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 5: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.38 \cdot 10^{-90} \lor \neg \left(x \leq 0.00096\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.38e-90) (not (<= x 0.00096)))
   (fabs (* x (/ (- 1.0 z) y)))
   (fabs (/ (+ x 4.0) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.38e-90) || !(x <= 0.00096)) {
		tmp = fabs((x * ((1.0 - z) / y)));
	} else {
		tmp = fabs(((x + 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 <= (-1.38d-90)) .or. (.not. (x <= 0.00096d0))) then
        tmp = abs((x * ((1.0d0 - z) / y)))
    else
        tmp = abs(((x + 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 <= -1.38e-90) || !(x <= 0.00096)) {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	} else {
		tmp = Math.abs(((x + 4.0) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (x <= -1.38e-90) or not (x <= 0.00096):
		tmp = math.fabs((x * ((1.0 - z) / y)))
	else:
		tmp = math.fabs(((x + 4.0) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.38e-90) || !(x <= 0.00096))
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y)));
	else
		tmp = abs(Float64(Float64(x + 4.0) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.38e-90) || ~((x <= 0.00096)))
		tmp = abs((x * ((1.0 - z) / y)));
	else
		tmp = abs(((x + 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, -1.38e-90], N[Not[LessEqual[x, 0.00096]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.37999999999999992e-90 or 9.60000000000000024e-4 < x

   1. Initial program 86.0%

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

      1. associate-*l/ 81.6%

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

      2. associate-*r/ 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-lft-in 89.6%

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

      4. associate-+r+ 89.6%

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

      5. distribute-rgt-in 89.7%

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

      6. associate-*l/ 89.8%

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

      7. *-lft-identity 89.8%

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

      8. +-commutative 89.8%

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

      9. distribute-rgt-neg-out 89.8%

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

      10. sub-neg 89.8%

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

      11. associate-*r/ 81.6%

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

      12. div-sub 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in x around inf 85.6%

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

      1. *-commutative 85.6%

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

      2. associate-*l/ 93.7%

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

      3. div-sub 93.7%

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

      4. *-commutative 93.7%

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

      5. div-sub 93.7%

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

   9. Simplified 93.7%

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

   if -1.37999999999999992e-90 < x < 9.60000000000000024e-4

   1. Initial program 97.3%

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

      1. associate-*l/ 99.9%

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

      2. associate-*r/ 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in z around 0 81.2%

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

      1. +-commutative 81.2%

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

      2. *-rgt-identity 81.2%

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

      3. associate-*r/ 81.2%

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

      4. distribute-rgt-in 81.2%

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

      5. associate-*l/ 81.3%

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

      6. *-lft-identity 81.3%

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

   6. Simplified 81.3%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.38 \cdot 10^{-90} \lor \neg \left(x \leq 0.00096\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \end{array} \]

Alternative 6: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (<= x 5e+14)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (* x (/ (- 1.0 z) y)))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= 5e+14) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((x * ((1.0 - 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 <= 5d+14) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs((x * ((1.0d0 - 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 <= 5e+14) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if x <= 5e+14:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((x * ((1.0 - z) / y)))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if (x <= 5e+14)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 5e+14)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((x * ((1.0 - z) / y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_, z_] := If[LessEqual[x, 5e+14], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e14

   1. Initial program 92.5%

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

      1. associate-*l/ 93.8%

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

      2. associate-*r/ 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in x around 0 94.1%

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

      1. sub-neg 94.1%

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

      2. +-commutative 94.1%

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

      3. distribute-lft-in 90.0%

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

      4. associate-+r+ 90.0%

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

      5. distribute-rgt-in 90.0%

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

      6. associate-*l/ 90.0%

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

      7. *-lft-identity 90.0%

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

      8. +-commutative 90.0%

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

      9. distribute-rgt-neg-out 90.0%

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

      10. sub-neg 90.0%

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

      11. associate-*r/ 93.8%

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

      12. div-sub 97.9%

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

   6. Simplified 97.9%

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

   if 5e14 < x

   1. Initial program 85.1%

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

      1. associate-*l/ 74.7%

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

      2. associate-*r/ 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-in 88.3%

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

      4. associate-+r+ 88.3%

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

      5. distribute-rgt-in 88.3%

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

      6. associate-*l/ 88.4%

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

      7. *-lft-identity 88.4%

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

      8. +-commutative 88.4%

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

      9. distribute-rgt-neg-out 88.4%

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

      10. sub-neg 88.4%

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

      11. associate-*r/ 74.7%

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

      12. div-sub 86.2%

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

   6. Simplified 86.2%

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

   7. Taylor expanded in x around inf 86.2%

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

      1. *-commutative 86.2%

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

      2. associate-*l/ 99.8%

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

      3. div-sub 99.8%

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

      4. *-commutative 99.8%

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

      5. div-sub 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \end{array} \]

Alternative 7: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+41} \lor \neg \left(z \leq 7.2 \cdot 10^{+27}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.4e+41) (not (<= z 7.2e+27)))
   (fabs (* z (/ x y)))
   (fabs (/ (+ x 4.0) y))))

C

y = abs(y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.4e+41) || !(z <= 7.2e+27)) {
		tmp = fabs((z * (x / y)));
	} else {
		tmp = fabs(((x + 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 ((z <= (-3.4d+41)) .or. (.not. (z <= 7.2d+27))) then
        tmp = abs((z * (x / y)))
    else
        tmp = abs(((x + 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 ((z <= -3.4e+41) || !(z <= 7.2e+27)) {
		tmp = Math.abs((z * (x / y)));
	} else {
		tmp = Math.abs(((x + 4.0) / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y, z):
	tmp = 0
	if (z <= -3.4e+41) or not (z <= 7.2e+27):
		tmp = math.fabs((z * (x / y)))
	else:
		tmp = math.fabs(((x + 4.0) / y))
	return tmp

Julia

y = abs(y)
function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.4e+41) || !(z <= 7.2e+27))
		tmp = abs(Float64(z * Float64(x / y)));
	else
		tmp = abs(Float64(Float64(x + 4.0) / y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.4e+41) || ~((z <= 7.2e+27)))
		tmp = abs((z * (x / y)));
	else
		tmp = abs(((x + 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[z, -3.4e+41], N[Not[LessEqual[z, 7.2e+27]], $MachinePrecision]], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.39999999999999998e41 or 7.19999999999999966e27 < z

   1. Initial program 90.7%

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

      1. associate-*l/ 83.9%

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

      2. associate-*r/ 85.6%

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

   3. Simplified 85.6%

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

   4. Taylor expanded in z around inf 69.2%

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

      1. associate-*r/ 69.2%

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

      2. neg-mul-1 69.2%

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

      3. distribute-lft-neg-in 69.2%

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

      4. associate-*l/ 76.2%

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

      5. *-commutative 76.2%

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

   6. Simplified 76.2%

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

      1. add-sqr-sqrt 33.1%

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

      2. sqrt-unprod 64.2%

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

      3. distribute-frac-neg 64.2%

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

      4. distribute-frac-neg 64.2%

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

      5. sqr-neg 64.2%

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

      6. sqrt-unprod 43.4%

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

      7. add-sqr-sqrt 76.2%

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

      8. clear-num 75.0%

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

      9. div-inv 75.1%

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

   8. Applied egg-rr 75.1%

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

      1. associate-/r/ 74.5%

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

      2. *-commutative 74.5%

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

      3. add-sqr-sqrt 33.6%

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

      4. sqrt-unprod 60.0%

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

      5. sqr-neg 60.0%

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

      6. sqrt-unprod 40.6%

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

      7. add-sqr-sqrt 74.5%

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

      8. associate-*r/ 69.2%

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

      9. associate-*l/ 76.2%

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

      10. add-sqr-sqrt 40.4%

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

      11. sqrt-unprod 60.0%

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

      12. sqr-neg 60.0%

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

      13. sqrt-unprod 35.5%

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

      14. add-sqr-sqrt 76.2%

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

   10. Applied egg-rr 76.2%

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

   if -3.39999999999999998e41 < z < 7.19999999999999966e27

   1. Initial program 90.8%

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

      1. associate-*l/ 93.6%

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

      2. associate-*r/ 93.0%

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

   3. Simplified 93.0%

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

   4. Taylor expanded in z around 0 96.2%

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

      1. +-commutative 96.2%

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

      2. *-rgt-identity 96.2%

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

      3. associate-*r/ 96.1%

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

      4. distribute-rgt-in 96.1%

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

      5. associate-*l/ 96.3%

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

      6. *-lft-identity 96.3%

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

   6. Simplified 96.3%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+41} \lor \neg \left(z \leq 7.2 \cdot 10^{+27}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \end{array} \]

Alternative 8: 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 84.4%

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

      1. associate-*l/ 77.8%

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

      2. associate-*r/ 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in x around inf 86.9%

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

   5. Taylor expanded in z around 0 65.4%

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

   if -1.5 < x < 4

   1. Initial program 96.5%

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

      1. associate-*l/ 99.8%

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

      2. associate-*r/ 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in x around 0 71.8%

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

4. Final simplification 68.7%

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

Alternative 9: 40.6% 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 90.7%

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

   1. associate-*l/ 89.2%

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

   2. associate-*r/ 89.6%

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

3. Simplified 89.6%

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

4. Taylor expanded in x around 0 39.8%

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

5. Final simplification 39.8%

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

Reproduce

herbie shell --seed 2023285 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))