fabs fraction 1

Percentage Accurate: 91.6% → 99.8%

Time: 7.0s

Alternatives: 10

Speedup: 1.0×

• 21.4% 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: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y\_m} - x \cdot \frac{z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= y_m 2e-26)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (- (/ (+ x 4.0) y_m) (* x (/ z y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2e-26) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs((((x + 4.0) / y_m) - (x * (z / y_m))));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 2d-26) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs((((x + 4.0d0) / y_m) - (x * (z / y_m))))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2e-26) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((((x + 4.0) / y_m) - (x * (z / y_m))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if y_m <= 2e-26:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs((((x + 4.0) / y_m) - (x * (z / y_m))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (y_m <= 2e-26)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) / y_m) - Float64(x * Float64(z / y_m))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (y_m <= 2e-26)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs((((x + 4.0) / y_m) - (x * (z / y_m))));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[y$95$m, 2e-26], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y$95$m), $MachinePrecision] - N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.0000000000000001e-26

   1. Initial program 88.7%

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

      1. associate-*l/ 90.7%

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

      2. sub-div 96.4%

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

   4. Applied egg-rr 96.4%

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

   if 2.0000000000000001e-26 < y

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0 94.1%

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

      1. associate-*r/ 99.9%

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

   5. Simplified 99.9%

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

Alternative 2: 69.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y\_m}\right|\\ t_1 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{+240}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{+198}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{+117}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-37} \lor \neg \left(x \leq 6.5 \cdot 10^{-28}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ x y_m))))
   (if (<= x -2.3e+240)
     t_0
     (if (<= x -7.8e+198)
       t_1
       (if (<= x -4.5e+117)
         (fabs (* x (/ z y_m)))
         (if (<= x -6.8e+75)
           t_1
           (if (or (<= x -4.5e-37) (not (<= x 6.5e-28)))
             t_0
             (fabs (/ 4.0 y_m)))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z * (x / y_m)));
	double t_1 = fabs((x / y_m));
	double tmp;
	if (x <= -2.3e+240) {
		tmp = t_0;
	} else if (x <= -7.8e+198) {
		tmp = t_1;
	} else if (x <= -4.5e+117) {
		tmp = fabs((x * (z / y_m)));
	} else if (x <= -6.8e+75) {
		tmp = t_1;
	} else if ((x <= -4.5e-37) || !(x <= 6.5e-28)) {
		tmp = t_0;
	} else {
		tmp = fabs((4.0 / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    t_1 = abs((x / y_m))
    if (x <= (-2.3d+240)) then
        tmp = t_0
    else if (x <= (-7.8d+198)) then
        tmp = t_1
    else if (x <= (-4.5d+117)) then
        tmp = abs((x * (z / y_m)))
    else if (x <= (-6.8d+75)) then
        tmp = t_1
    else if ((x <= (-4.5d-37)) .or. (.not. (x <= 6.5d-28))) then
        tmp = t_0
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((z * (x / y_m)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (x <= -2.3e+240) {
		tmp = t_0;
	} else if (x <= -7.8e+198) {
		tmp = t_1;
	} else if (x <= -4.5e+117) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (x <= -6.8e+75) {
		tmp = t_1;
	} else if ((x <= -4.5e-37) || !(x <= 6.5e-28)) {
		tmp = t_0;
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z * (x / y_m)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if x <= -2.3e+240:
		tmp = t_0
	elif x <= -7.8e+198:
		tmp = t_1
	elif x <= -4.5e+117:
		tmp = math.fabs((x * (z / y_m)))
	elif x <= -6.8e+75:
		tmp = t_1
	elif (x <= -4.5e-37) or not (x <= 6.5e-28):
		tmp = t_0
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z * Float64(x / y_m)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -2.3e+240)
		tmp = t_0;
	elseif (x <= -7.8e+198)
		tmp = t_1;
	elseif (x <= -4.5e+117)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (x <= -6.8e+75)
		tmp = t_1;
	elseif ((x <= -4.5e-37) || !(x <= 6.5e-28))
		tmp = t_0;
	else
		tmp = abs(Float64(4.0 / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((z * (x / y_m)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -2.3e+240)
		tmp = t_0;
	elseif (x <= -7.8e+198)
		tmp = t_1;
	elseif (x <= -4.5e+117)
		tmp = abs((x * (z / y_m)));
	elseif (x <= -6.8e+75)
		tmp = t_1;
	elseif ((x <= -4.5e-37) || ~((x <= 6.5e-28)))
		tmp = t_0;
	else
		tmp = abs((4.0 / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.3e+240], t$95$0, If[LessEqual[x, -7.8e+198], t$95$1, If[LessEqual[x, -4.5e+117], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -6.8e+75], t$95$1, If[Or[LessEqual[x, -4.5e-37], N[Not[LessEqual[x, 6.5e-28]], $MachinePrecision]], t$95$0, N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.30000000000000001e240 or -6.80000000000000022e75 < x < -4.5000000000000004e-37 or 6.50000000000000043e-28 < x

   1. Initial program 84.9%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in z around inf 54.5%

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

      1. mul-1-neg 54.5%

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

      2. associate-*r/ 63.2%

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

      3. *-commutative 63.2%

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

      4. distribute-rgt-neg-in 63.2%

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

   7. Simplified 63.2%

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

      1. *-commutative 63.2%

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

      2. clear-num 63.1%

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

      3. un-div-inv 63.2%

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

      4. add-sqr-sqrt 22.2%

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

      5. sqrt-unprod 60.2%

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

      6. sqr-neg 60.2%

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

      7. sqrt-unprod 40.9%

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

      8. add-sqr-sqrt 63.2%

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

   9. Applied egg-rr 63.2%

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

       1. associate-/r/ 69.7%

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

       2. *-commutative 69.7%

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

   11. Simplified 69.7%

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

   if -2.30000000000000001e240 < x < -7.8e198 or -4.5e117 < x < -6.80000000000000022e75

   1. Initial program 95.8%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in x around inf 96.1%

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

      1. associate-*r/ 96.1%

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

      2. *-commutative 96.1%

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

      3. associate-*r* 96.1%

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

      4. sub-neg 96.1%

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

      5. metadata-eval 96.1%

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

      6. distribute-lft-in 96.1%

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

      7. neg-mul-1 96.1%

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

      8. metadata-eval 96.1%

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

      9. +-commutative 96.1%

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

      10. neg-mul-1 96.1%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in z around 0 88.4%

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

   if -7.8e198 < x < -4.5e117

   1. Initial program 74.6%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in z around inf 75.9%

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

      1. associate-*r/ 75.9%

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

      2. neg-mul-1 75.9%

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

      3. distribute-rgt-neg-in 75.9%

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

   7. Simplified 75.9%

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

      1. distribute-rgt-neg-out 75.9%

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

      2. distribute-lft-neg-in 75.9%

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

      3. associate-*r/ 99.8%

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

      4. *-commutative 99.8%

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

      5. add-sqr-sqrt 99.4%

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

      6. sqrt-unprod 88.0%

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

      7. sqr-neg 88.0%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -4.5000000000000004e-37 < x < 6.50000000000000043e-28

   1. Initial program 94.1%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 79.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+240}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{+198}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{+117}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{+75}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-37} \lor \neg \left(x \leq 6.5 \cdot 10^{-28}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y\_m}\right|\\ t_1 := \left|\frac{x}{y\_m}\right|\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+245}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{+199}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{+117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-35} \lor \neg \left(x \leq 3 \cdot 10^{-26}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y_m)))) (t_1 (fabs (/ x y_m))))
   (if (<= x -1.6e+245)
     t_0
     (if (<= x -1.05e+199)
       t_1
       (if (<= x -6.2e+117)
         t_0
         (if (<= x -1.45e+76)
           t_1
           (if (or (<= x -4.9e-35) (not (<= x 3e-26)))
             t_0
             (fabs (/ 4.0 y_m)))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double t_0 = fabs((z * (x / y_m)));
	double t_1 = fabs((x / y_m));
	double tmp;
	if (x <= -1.6e+245) {
		tmp = t_0;
	} else if (x <= -1.05e+199) {
		tmp = t_1;
	} else if (x <= -6.2e+117) {
		tmp = t_0;
	} else if (x <= -1.45e+76) {
		tmp = t_1;
	} else if ((x <= -4.9e-35) || !(x <= 3e-26)) {
		tmp = t_0;
	} else {
		tmp = fabs((4.0 / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs((z * (x / y_m)))
    t_1 = abs((x / y_m))
    if (x <= (-1.6d+245)) then
        tmp = t_0
    else if (x <= (-1.05d+199)) then
        tmp = t_1
    else if (x <= (-6.2d+117)) then
        tmp = t_0
    else if (x <= (-1.45d+76)) then
        tmp = t_1
    else if ((x <= (-4.9d-35)) .or. (.not. (x <= 3d-26))) then
        tmp = t_0
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double t_0 = Math.abs((z * (x / y_m)));
	double t_1 = Math.abs((x / y_m));
	double tmp;
	if (x <= -1.6e+245) {
		tmp = t_0;
	} else if (x <= -1.05e+199) {
		tmp = t_1;
	} else if (x <= -6.2e+117) {
		tmp = t_0;
	} else if (x <= -1.45e+76) {
		tmp = t_1;
	} else if ((x <= -4.9e-35) || !(x <= 3e-26)) {
		tmp = t_0;
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	t_0 = math.fabs((z * (x / y_m)))
	t_1 = math.fabs((x / y_m))
	tmp = 0
	if x <= -1.6e+245:
		tmp = t_0
	elif x <= -1.05e+199:
		tmp = t_1
	elif x <= -6.2e+117:
		tmp = t_0
	elif x <= -1.45e+76:
		tmp = t_1
	elif (x <= -4.9e-35) or not (x <= 3e-26):
		tmp = t_0
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	t_0 = abs(Float64(z * Float64(x / y_m)))
	t_1 = abs(Float64(x / y_m))
	tmp = 0.0
	if (x <= -1.6e+245)
		tmp = t_0;
	elseif (x <= -1.05e+199)
		tmp = t_1;
	elseif (x <= -6.2e+117)
		tmp = t_0;
	elseif (x <= -1.45e+76)
		tmp = t_1;
	elseif ((x <= -4.9e-35) || !(x <= 3e-26))
		tmp = t_0;
	else
		tmp = abs(Float64(4.0 / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	t_0 = abs((z * (x / y_m)));
	t_1 = abs((x / y_m));
	tmp = 0.0;
	if (x <= -1.6e+245)
		tmp = t_0;
	elseif (x <= -1.05e+199)
		tmp = t_1;
	elseif (x <= -6.2e+117)
		tmp = t_0;
	elseif (x <= -1.45e+76)
		tmp = t_1;
	elseif ((x <= -4.9e-35) || ~((x <= 3e-26)))
		tmp = t_0;
	else
		tmp = abs((4.0 / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.6e+245], t$95$0, If[LessEqual[x, -1.05e+199], t$95$1, If[LessEqual[x, -6.2e+117], t$95$0, If[LessEqual[x, -1.45e+76], t$95$1, If[Or[LessEqual[x, -4.9e-35], N[Not[LessEqual[x, 3e-26]], $MachinePrecision]], t$95$0, N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.60000000000000012e245 or -1.05e199 < x < -6.1999999999999995e117 or -1.4500000000000001e76 < x < -4.9000000000000005e-35 or 3.00000000000000012e-26 < x

   1. Initial program 84.0%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in z around inf 56.3%

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

      1. mul-1-neg 56.3%

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

      2. associate-*r/ 66.3%

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

      3. *-commutative 66.3%

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

      4. distribute-rgt-neg-in 66.3%

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

   7. Simplified 66.3%

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

      1. *-commutative 66.3%

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

      2. clear-num 66.1%

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

      3. un-div-inv 66.3%

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

      4. add-sqr-sqrt 28.8%

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

      5. sqrt-unprod 62.5%

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

      6. sqr-neg 62.5%

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

      7. sqrt-unprod 37.4%

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

      8. add-sqr-sqrt 66.3%

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

   9. Applied egg-rr 66.3%

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

       1. associate-/r/ 72.2%

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

       2. *-commutative 72.2%

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

   11. Simplified 72.2%

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

   if -1.60000000000000012e245 < x < -1.05e199 or -6.1999999999999995e117 < x < -1.4500000000000001e76

   1. Initial program 95.8%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in x around inf 96.1%

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

      1. associate-*r/ 96.1%

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

      2. *-commutative 96.1%

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

      3. associate-*r* 96.1%

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

      4. sub-neg 96.1%

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

      5. metadata-eval 96.1%

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

      6. distribute-lft-in 96.1%

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

      7. neg-mul-1 96.1%

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

      8. metadata-eval 96.1%

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

      9. +-commutative 96.1%

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

      10. neg-mul-1 96.1%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in z around 0 88.4%

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

   if -4.9000000000000005e-35 < x < 3.00000000000000012e-26

   1. Initial program 94.1%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 79.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+245}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{+199}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{+117}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+76}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-35} \lor \neg \left(x \leq 3 \cdot 10^{-26}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+52}:\\ \;\;\;\;\left|x \cdot \frac{z}{y\_m}\right|\\ \mathbf{elif}\;z \leq 1.42 \cdot 10^{+75}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+175}:\\ \;\;\;\;\left|z \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;z \leq 10^{+192}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+288}:\\ \;\;\;\;\left|\frac{x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y\_m}{x}}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= z -1.2e+52)
   (fabs (* x (/ z y_m)))
   (if (<= z 1.42e+75)
     (fabs (/ (- -4.0 x) y_m))
     (if (<= z 3.9e+175)
       (fabs (* z (/ x y_m)))
       (if (<= z 1e+192)
         (fabs (/ 4.0 y_m))
         (if (<= z 1.7e+288)
           (fabs (/ (* x z) y_m))
           (fabs (/ z (/ y_m x)))))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.2e+52) {
		tmp = fabs((x * (z / y_m)));
	} else if (z <= 1.42e+75) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else if (z <= 3.9e+175) {
		tmp = fabs((z * (x / y_m)));
	} else if (z <= 1e+192) {
		tmp = fabs((4.0 / y_m));
	} else if (z <= 1.7e+288) {
		tmp = fabs(((x * z) / y_m));
	} else {
		tmp = fabs((z / (y_m / x)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.2d+52)) then
        tmp = abs((x * (z / y_m)))
    else if (z <= 1.42d+75) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else if (z <= 3.9d+175) then
        tmp = abs((z * (x / y_m)))
    else if (z <= 1d+192) then
        tmp = abs((4.0d0 / y_m))
    else if (z <= 1.7d+288) then
        tmp = abs(((x * z) / y_m))
    else
        tmp = abs((z / (y_m / x)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (z <= -1.2e+52) {
		tmp = Math.abs((x * (z / y_m)));
	} else if (z <= 1.42e+75) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else if (z <= 3.9e+175) {
		tmp = Math.abs((z * (x / y_m)));
	} else if (z <= 1e+192) {
		tmp = Math.abs((4.0 / y_m));
	} else if (z <= 1.7e+288) {
		tmp = Math.abs(((x * z) / y_m));
	} else {
		tmp = Math.abs((z / (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if z <= -1.2e+52:
		tmp = math.fabs((x * (z / y_m)))
	elif z <= 1.42e+75:
		tmp = math.fabs(((-4.0 - x) / y_m))
	elif z <= 3.9e+175:
		tmp = math.fabs((z * (x / y_m)))
	elif z <= 1e+192:
		tmp = math.fabs((4.0 / y_m))
	elif z <= 1.7e+288:
		tmp = math.fabs(((x * z) / y_m))
	else:
		tmp = math.fabs((z / (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (z <= -1.2e+52)
		tmp = abs(Float64(x * Float64(z / y_m)));
	elseif (z <= 1.42e+75)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	elseif (z <= 3.9e+175)
		tmp = abs(Float64(z * Float64(x / y_m)));
	elseif (z <= 1e+192)
		tmp = abs(Float64(4.0 / y_m));
	elseif (z <= 1.7e+288)
		tmp = abs(Float64(Float64(x * z) / y_m));
	else
		tmp = abs(Float64(z / Float64(y_m / x)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (z <= -1.2e+52)
		tmp = abs((x * (z / y_m)));
	elseif (z <= 1.42e+75)
		tmp = abs(((-4.0 - x) / y_m));
	elseif (z <= 3.9e+175)
		tmp = abs((z * (x / y_m)));
	elseif (z <= 1e+192)
		tmp = abs((4.0 / y_m));
	elseif (z <= 1.7e+288)
		tmp = abs(((x * z) / y_m));
	else
		tmp = abs((z / (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[z, -1.2e+52], N[Abs[N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.42e+75], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3.9e+175], N[Abs[N[(z * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1e+192], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.7e+288], N[Abs[N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if z < -1.2e52

   1. Initial program 90.3%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in z around inf 67.4%

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

      1. associate-*r/ 67.4%

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

      2. neg-mul-1 67.4%

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

      3. distribute-rgt-neg-in 67.4%

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

   7. Simplified 67.4%

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

      1. distribute-rgt-neg-out 67.4%

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

      2. distribute-lft-neg-in 67.4%

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

      3. associate-*r/ 77.7%

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

      4. *-commutative 77.7%

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

      5. add-sqr-sqrt 35.5%

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

      6. sqrt-unprod 60.1%

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

      7. sqr-neg 60.1%

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

      8. sqrt-unprod 42.0%

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

      9. add-sqr-sqrt 77.7%

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

   9. Applied egg-rr 77.7%

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

   if -1.2e52 < z < 1.42e75

   1. Initial program 93.3%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 93.9%

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

      1. +-commutative 93.9%

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

      2. rem-square-sqrt 42.3%

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

      3. fabs-sqr 42.3%

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

      4. rem-square-sqrt 93.9%

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

      5. fabs-neg 93.9%

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

      6. distribute-neg-frac 93.9%

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

      7. distribute-neg-in 93.9%

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

      8. metadata-eval 93.9%

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

      9. +-commutative 93.9%

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

      10. sub-neg 93.9%

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

      11. rem-square-sqrt 51.1%

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

      12. fabs-sqr 51.1%

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

      13. rem-square-sqrt 93.9%

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

   7. Simplified 93.9%

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

   if 1.42e75 < z < 3.89999999999999972e175

   1. Initial program 79.9%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in z around inf 65.6%

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

      1. mul-1-neg 65.6%

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

      2. associate-*r/ 71.1%

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

      3. *-commutative 71.1%

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

      4. distribute-rgt-neg-in 71.1%

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

   7. Simplified 71.1%

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

      1. *-commutative 71.1%

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

      2. clear-num 71.0%

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

      3. un-div-inv 71.3%

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

      4. add-sqr-sqrt 12.2%

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

      5. sqrt-unprod 55.4%

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

      6. sqr-neg 55.4%

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

      7. sqrt-unprod 58.8%

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

      8. add-sqr-sqrt 71.3%

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

   9. Applied egg-rr 71.3%

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

       1. associate-/r/ 75.6%

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

       2. *-commutative 75.6%

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

   11. Simplified 75.6%

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

   if 3.89999999999999972e175 < z < 1.00000000000000004e192

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 1.00000000000000004e192 < z < 1.70000000000000009e288

   1. Initial program 71.2%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 76.7%

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

      1. mul-1-neg 76.7%

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

      2. associate-*r/ 72.7%

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

      3. *-commutative 72.7%

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

      4. distribute-rgt-neg-in 72.7%

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

   7. Simplified 72.7%

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

      1. *-commutative 72.7%

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

      2. add-sqr-sqrt 47.7%

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

      3. sqrt-unprod 67.9%

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

      4. sqr-neg 67.9%

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

      5. sqrt-unprod 24.9%

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

      6. add-sqr-sqrt 72.7%

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

      7. associate-/l* 76.7%

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

   9. Applied egg-rr 76.7%

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

   if 1.70000000000000009e288 < z

   1. Initial program 99.5%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in z around inf 89.2%

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

      1. mul-1-neg 89.2%

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

      2. associate-*r/ 89.4%

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

      3. *-commutative 89.4%

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

      4. distribute-rgt-neg-in 89.4%

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

   7. Simplified 89.4%

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

      1. *-commutative 89.4%

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

      2. clear-num 89.1%

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

      3. un-div-inv 89.1%

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

      4. add-sqr-sqrt 55.8%

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

      5. sqrt-unprod 67.3%

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

      6. sqr-neg 67.3%

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

      7. sqrt-unprod 33.2%

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

      8. add-sqr-sqrt 89.1%

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

   9. Applied egg-rr 89.1%

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

       1. associate-/r/ 99.5%

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

       2. *-commutative 99.5%

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

   11. Simplified 99.5%

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

       1. clear-num 99.5%

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

       2. div-inv 100.0%

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

   13. Applied egg-rr 100.0%

       \[\leadsto \left|\color{blue}{\frac{z}{\frac{y}{x}}}\right| \]
3. Recombined 6 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+52}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 1.42 \cdot 10^{+75}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+175}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 10^{+192}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+288}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -530000000 \lor \neg \left(x \leq 1.75 \cdot 10^{-26}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -530000000.0) (not (<= x 1.75e-26)))
   (fabs (* x (/ (- 1.0 z) y_m)))
   (fabs (/ (- -4.0 x) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -530000000.0) || !(x <= 1.75e-26)) {
		tmp = fabs((x * ((1.0 - z) / y_m)));
	} else {
		tmp = fabs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-530000000.0d0)) .or. (.not. (x <= 1.75d-26))) then
        tmp = abs((x * ((1.0d0 - z) / y_m)))
    else
        tmp = abs((((-4.0d0) - x) / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -530000000.0) || !(x <= 1.75e-26)) {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -530000000.0) or not (x <= 1.75e-26):
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	else:
		tmp = math.fabs(((-4.0 - x) / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -530000000.0) || !(x <= 1.75e-26))
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -530000000.0) || ~((x <= 1.75e-26)))
		tmp = abs((x * ((1.0 - z) / y_m)));
	else
		tmp = abs(((-4.0 - x) / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -530000000.0], N[Not[LessEqual[x, 1.75e-26]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.3e8 or 1.74999999999999992e-26 < x

   1. Initial program 85.6%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in x around inf 89.1%

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

      1. mul-1-neg 89.1%

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

      2. associate-/l* 98.3%

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

      3. distribute-rgt-neg-in 98.3%

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

      4. mul-1-neg 98.3%

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

      5. associate-*r/ 98.3%

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

      6. sub-neg 98.3%

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

      7. metadata-eval 98.3%

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

      8. distribute-lft-in 98.3%

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

      9. neg-mul-1 98.3%

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

      10. metadata-eval 98.3%

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

      11. +-commutative 98.3%

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

      12. unsub-neg 98.3%

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

   7. Simplified 98.3%

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

   if -5.3e8 < x < 1.74999999999999992e-26

   1. Initial program 94.3%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 78.9%

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

      1. +-commutative 78.9%

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

      2. rem-square-sqrt 36.6%

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

      3. fabs-sqr 36.6%

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

      4. rem-square-sqrt 78.9%

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

      5. fabs-neg 78.9%

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

      6. distribute-neg-frac 78.9%

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

      7. distribute-neg-in 78.9%

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

      8. metadata-eval 78.9%

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

      9. +-commutative 78.9%

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

      10. sub-neg 78.9%

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

      11. rem-square-sqrt 41.8%

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

      12. fabs-sqr 41.8%

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

      13. rem-square-sqrt 78.9%

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

   7. Simplified 78.9%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -530000000 \lor \neg \left(x \leq 1.75 \cdot 10^{-26}\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -530000000:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-24}:\\ \;\;\;\;\left|\frac{-4}{y\_m} - \frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -530000000.0)
   (fabs (* (- 1.0 z) (/ x y_m)))
   (if (<= x 2.6e-24)
     (fabs (- (/ -4.0 y_m) (/ x y_m)))
     (fabs (* x (/ (- 1.0 z) y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -530000000.0) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else if (x <= 2.6e-24) {
		tmp = fabs(((-4.0 / y_m) - (x / y_m)));
	} else {
		tmp = fabs((x * ((1.0 - z) / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-530000000.0d0)) then
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    else if (x <= 2.6d-24) then
        tmp = abs((((-4.0d0) / y_m) - (x / y_m)))
    else
        tmp = abs((x * ((1.0d0 - z) / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -530000000.0) {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	} else if (x <= 2.6e-24) {
		tmp = Math.abs(((-4.0 / y_m) - (x / y_m)));
	} else {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -530000000.0:
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	elif x <= 2.6e-24:
		tmp = math.fabs(((-4.0 / y_m) - (x / y_m)))
	else:
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -530000000.0)
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	elseif (x <= 2.6e-24)
		tmp = abs(Float64(Float64(-4.0 / y_m) - Float64(x / y_m)));
	else
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -530000000.0)
		tmp = abs(((1.0 - z) * (x / y_m)));
	elseif (x <= 2.6e-24)
		tmp = abs(((-4.0 / y_m) - (x / y_m)));
	else
		tmp = abs((x * ((1.0 - z) / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -530000000.0], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.6e-24], N[Abs[N[(N[(-4.0 / y$95$m), $MachinePrecision] - N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.3e8

   1. Initial program 84.8%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in x around inf 92.8%

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

      1. associate-*r/ 92.8%

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

      2. *-commutative 92.8%

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

      3. associate-*r* 92.8%

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

      4. sub-neg 92.8%

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

      5. metadata-eval 92.8%

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

      6. distribute-lft-in 92.8%

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

      7. neg-mul-1 92.8%

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

      8. metadata-eval 92.8%

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

      9. +-commutative 92.8%

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

      10. neg-mul-1 92.8%

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

      11. associate-/l* 99.9%

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

      12. neg-mul-1 99.9%

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

      13. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -5.3e8 < x < 2.6e-24

   1. Initial program 94.3%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 78.9%

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

      1. +-commutative 78.9%

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

      2. rem-square-sqrt 36.6%

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

      3. fabs-sqr 36.6%

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

      4. rem-square-sqrt 78.9%

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

      5. fabs-neg 78.9%

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

      6. distribute-neg-frac 78.9%

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

      7. distribute-neg-in 78.9%

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

      8. metadata-eval 78.9%

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

      9. +-commutative 78.9%

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

      10. sub-neg 78.9%

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

      11. rem-square-sqrt 41.8%

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

      12. fabs-sqr 41.8%

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

      13. rem-square-sqrt 78.9%

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

   7. Simplified 78.9%

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

      1. div-sub 78.9%

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

   9. Applied egg-rr 78.9%

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

   if 2.6e-24 < x

   1. Initial program 86.2%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in x around inf 85.8%

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

      1. mul-1-neg 85.8%

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

      2. associate-/l* 96.9%

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

      3. distribute-rgt-neg-in 96.9%

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

      4. mul-1-neg 96.9%

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

      5. associate-*r/ 96.9%

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

      6. sub-neg 96.9%

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

      7. metadata-eval 96.9%

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

      8. distribute-lft-in 96.9%

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

      9. neg-mul-1 96.9%

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

      10. metadata-eval 96.9%

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

      11. +-commutative 96.9%

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

      12. unsub-neg 96.9%

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

   7. Simplified 96.9%

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

Alternative 7: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -530000000:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y\_m}\right|\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-26}:\\ \;\;\;\;\left|\frac{-4 - x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x -530000000.0)
   (fabs (* (- 1.0 z) (/ x y_m)))
   (if (<= x 4.3e-26)
     (fabs (/ (- -4.0 x) y_m))
     (fabs (* x (/ (- 1.0 z) y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -530000000.0) {
		tmp = fabs(((1.0 - z) * (x / y_m)));
	} else if (x <= 4.3e-26) {
		tmp = fabs(((-4.0 - x) / y_m));
	} else {
		tmp = fabs((x * ((1.0 - z) / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-530000000.0d0)) then
        tmp = abs(((1.0d0 - z) * (x / y_m)))
    else if (x <= 4.3d-26) then
        tmp = abs((((-4.0d0) - x) / y_m))
    else
        tmp = abs((x * ((1.0d0 - z) / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= -530000000.0) {
		tmp = Math.abs(((1.0 - z) * (x / y_m)));
	} else if (x <= 4.3e-26) {
		tmp = Math.abs(((-4.0 - x) / y_m));
	} else {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= -530000000.0:
		tmp = math.fabs(((1.0 - z) * (x / y_m)))
	elif x <= 4.3e-26:
		tmp = math.fabs(((-4.0 - x) / y_m))
	else:
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= -530000000.0)
		tmp = abs(Float64(Float64(1.0 - z) * Float64(x / y_m)));
	elseif (x <= 4.3e-26)
		tmp = abs(Float64(Float64(-4.0 - x) / y_m));
	else
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= -530000000.0)
		tmp = abs(((1.0 - z) * (x / y_m)));
	elseif (x <= 4.3e-26)
		tmp = abs(((-4.0 - x) / y_m));
	else
		tmp = abs((x * ((1.0 - z) / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, -530000000.0], N[Abs[N[(N[(1.0 - z), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.3e-26], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.3e8

   1. Initial program 84.8%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in x around inf 92.8%

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

      1. associate-*r/ 92.8%

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

      2. *-commutative 92.8%

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

      3. associate-*r* 92.8%

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

      4. sub-neg 92.8%

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

      5. metadata-eval 92.8%

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

      6. distribute-lft-in 92.8%

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

      7. neg-mul-1 92.8%

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

      8. metadata-eval 92.8%

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

      9. +-commutative 92.8%

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

      10. neg-mul-1 92.8%

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

      11. associate-/l* 99.9%

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

      12. neg-mul-1 99.9%

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

      13. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -5.3e8 < x < 4.29999999999999988e-26

   1. Initial program 94.3%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 78.9%

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

      1. +-commutative 78.9%

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

      2. rem-square-sqrt 36.6%

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

      3. fabs-sqr 36.6%

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

      4. rem-square-sqrt 78.9%

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

      5. fabs-neg 78.9%

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

      6. distribute-neg-frac 78.9%

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

      7. distribute-neg-in 78.9%

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

      8. metadata-eval 78.9%

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

      9. +-commutative 78.9%

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

      10. sub-neg 78.9%

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

      11. rem-square-sqrt 41.8%

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

      12. fabs-sqr 41.8%

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

      13. rem-square-sqrt 78.9%

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

   7. Simplified 78.9%

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

   if 4.29999999999999988e-26 < x

   1. Initial program 86.2%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in x around inf 85.8%

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

      1. mul-1-neg 85.8%

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

      2. associate-/l* 96.9%

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

      3. distribute-rgt-neg-in 96.9%

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

      4. mul-1-neg 96.9%

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

      5. associate-*r/ 96.9%

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

      6. sub-neg 96.9%

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

      7. metadata-eval 96.9%

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

      8. distribute-lft-in 96.9%

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

      9. neg-mul-1 96.9%

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

      10. metadata-eval 96.9%

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

      11. +-commutative 96.9%

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

      12. unsub-neg 96.9%

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

   7. Simplified 96.9%

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

Alternative 8: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+39}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (<= x 3.7e+39)
   (fabs (/ (- (+ x 4.0) (* x z)) y_m))
   (fabs (* x (/ (- 1.0 z) y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 3.7e+39) {
		tmp = fabs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = fabs((x * ((1.0 - z) / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 3.7d+39) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y_m))
    else
        tmp = abs((x * ((1.0d0 - z) / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if (x <= 3.7e+39) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y_m));
	} else {
		tmp = Math.abs((x * ((1.0 - z) / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if x <= 3.7e+39:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y_m))
	else:
		tmp = math.fabs((x * ((1.0 - z) / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if (x <= 3.7e+39)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y_m));
	else
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y_m)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if (x <= 3.7e+39)
		tmp = abs((((x + 4.0) - (x * z)) / y_m));
	else
		tmp = abs((x * ((1.0 - z) / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[LessEqual[x, 3.7e+39], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.70000000000000012e39

   1. Initial program 92.0%

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

      1. associate-*l/ 95.6%

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

      2. sub-div 98.0%

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

   4. Applied egg-rr 98.0%

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

   if 3.70000000000000012e39 < x

   1. Initial program 84.4%

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

   3. Simplified 87.1%

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

   5. Taylor expanded in x around inf 87.2%

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

      1. mul-1-neg 87.2%

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

      2. associate-/l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. mul-1-neg 99.8%

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

      5. associate-*r/ 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

      8. distribute-lft-in 99.8%

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

      9. neg-mul-1 99.8%

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

      10. metadata-eval 99.8%

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

      11. +-commutative 99.8%

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

      12. unsub-neg 99.8%

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

   7. Simplified 99.8%

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

Alternative 9: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y\_m}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y\_m}\right|\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 4.0))) (fabs (/ x y_m)) (fabs (/ 4.0 y_m))))

C

y_m = fabs(y);
double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.55) || !(x <= 4.0)) {
		tmp = fabs((x / y_m));
	} else {
		tmp = fabs((4.0 / y_m));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.55d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y_m))
    else
        tmp = abs((4.0d0 / y_m))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m, double z) {
	double tmp;
	if ((x <= -1.55) || !(x <= 4.0)) {
		tmp = Math.abs((x / y_m));
	} else {
		tmp = Math.abs((4.0 / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m, z):
	tmp = 0
	if (x <= -1.55) or not (x <= 4.0):
		tmp = math.fabs((x / y_m))
	else:
		tmp = math.fabs((4.0 / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m, z)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 4.0))
		tmp = abs(Float64(x / y_m));
	else
		tmp = abs(Float64(4.0 / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m, z)
	tmp = 0.0;
	if ((x <= -1.55) || ~((x <= 4.0)))
		tmp = abs((x / y_m));
	else
		tmp = abs((4.0 / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y$95$m), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 4 < x

   1. Initial program 85.4%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in x around inf 88.8%

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

      1. associate-*r/ 88.8%

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

      2. *-commutative 88.8%

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

      3. associate-*r* 88.8%

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

      4. sub-neg 88.8%

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

      5. metadata-eval 88.8%

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

      6. distribute-lft-in 88.8%

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

      7. neg-mul-1 88.8%

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

      8. metadata-eval 88.8%

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

      9. +-commutative 88.8%

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

      10. neg-mul-1 88.8%

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

      11. associate-/l* 98.0%

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

      12. neg-mul-1 98.0%

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

      13. unsub-neg 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in z around 0 61.7%

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

   if -1.55000000000000004 < x < 4

   1. Initial program 94.4%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 77.3%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.4% accurate, 1.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m z) :precision binary64 (fabs (/ 4.0 y_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_, z_] := N[Abs[N[(4.0 / y$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 90.5%

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

3. Simplified 95.8%

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

5. Taylor expanded in x around 0 46.1%

   \[\leadsto \left|\color{blue}{\frac{4}{y}}\right| \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024106 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))