fabs fraction 1

Percentage Accurate: 91.9% → 99.7%

Time: 8.8s

Alternatives: 11

Speedup: 1.0×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-65} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+299}\right):\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (- (/ (+ x 4.0) y) (* z (/ x y))))))
   (if (or (<= t_0 4e-65) (not (<= t_0 5e+299)))
     (fabs (/ (- (+ x 4.0) (* x z)) y))
     t_0)))

C

double code(double x, double y, double z) {
	double t_0 = fabs((((x + 4.0) / y) - (z * (x / y))));
	double tmp;
	if ((t_0 <= 4e-65) || !(t_0 <= 5e+299)) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((((x + 4.0d0) / y) - (z * (x / y))))
    if ((t_0 <= 4d-65) .or. (.not. (t_0 <= 5d+299))) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((((x + 4.0) / y) - (z * (x / y))));
	double tmp;
	if ((t_0 <= 4e-65) || !(t_0 <= 5e+299)) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((((x + 4.0) / y) - (z * (x / y))))
	tmp = 0
	if (t_0 <= 4e-65) or not (t_0 <= 5e+299):
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(z * Float64(x / y))))
	tmp = 0.0
	if ((t_0 <= 4e-65) || !(t_0 <= 5e+299))
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((((x + 4.0) / y) - (z * (x / y))));
	tmp = 0.0;
	if ((t_0 <= 4e-65) || ~((t_0 <= 5e+299)))
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t$95$0, 4e-65], N[Not[LessEqual[t$95$0, 5e+299]], $MachinePrecision]], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))) < 3.99999999999999969e-65 or 5.0000000000000003e299 < (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 83.3%

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

      1. associate-*l/ 90.9%

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

      2. sub-div 100.0%

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

   4. Applied egg-rr 100.0%

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

   if 3.99999999999999969e-65 < (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))) < 5.0000000000000003e299

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 4 \cdot 10^{-65} \lor \neg \left(\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 5 \cdot 10^{+299}\right):\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -15000000000:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 3.7:\\ \;\;\;\;\left|\frac{4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -15000000000.0)
   (fabs (* (/ x y) (- 1.0 z)))
   (if (<= x 3.7)
     (fabs (- (/ 4.0 y) (* x (/ z y))))
     (fabs (/ x (/ y (- 1.0 z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -15000000000.0) {
		tmp = fabs(((x / y) * (1.0 - z)));
	} else if (x <= 3.7) {
		tmp = fabs(((4.0 / y) - (x * (z / y))));
	} else {
		tmp = fabs((x / (y / (1.0 - z))));
	}
	return tmp;
}

Fortran

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 <= (-15000000000.0d0)) then
        tmp = abs(((x / y) * (1.0d0 - z)))
    else if (x <= 3.7d0) then
        tmp = abs(((4.0d0 / y) - (x * (z / y))))
    else
        tmp = abs((x / (y / (1.0d0 - z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -15000000000.0) {
		tmp = Math.abs(((x / y) * (1.0 - z)));
	} else if (x <= 3.7) {
		tmp = Math.abs(((4.0 / y) - (x * (z / y))));
	} else {
		tmp = Math.abs((x / (y / (1.0 - z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -15000000000.0:
		tmp = math.fabs(((x / y) * (1.0 - z)))
	elif x <= 3.7:
		tmp = math.fabs(((4.0 / y) - (x * (z / y))))
	else:
		tmp = math.fabs((x / (y / (1.0 - z))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -15000000000.0)
		tmp = abs(Float64(Float64(x / y) * Float64(1.0 - z)));
	elseif (x <= 3.7)
		tmp = abs(Float64(Float64(4.0 / y) - Float64(x * Float64(z / y))));
	else
		tmp = abs(Float64(x / Float64(y / Float64(1.0 - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -15000000000.0)
		tmp = abs(((x / y) * (1.0 - z)));
	elseif (x <= 3.7)
		tmp = abs(((4.0 / y) - (x * (z / y))));
	else
		tmp = abs((x / (y / (1.0 - z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -15000000000.0], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.7], N[Abs[N[(N[(4.0 / y), $MachinePrecision] - N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e10

   1. Initial program 89.8%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in x around inf 87.5%

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

      1. mul-1-neg 87.5%

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

      2. *-commutative 87.5%

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

      3. associate-/l* 99.8%

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

      4. distribute-lft-neg-in 99.8%

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

      5. neg-sub0 99.8%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 99.8%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 99.8%

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

      8. +-commutative 99.8%

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

      9. unsub-neg 99.8%

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

   7. Simplified 99.8%

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

   if -1.5e10 < x < 3.7000000000000002

   1. Initial program 96.2%

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

   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 -inf 83.4%

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

      1. mul-1-neg 83.4%

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

      2. distribute-rgt-neg-in 83.4%

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

      3. *-lft-identity 83.4%

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

      4. metadata-eval 83.4%

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

      5. cancel-sign-sub-inv 83.4%

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

      6. distribute-lft-out-- 83.4%

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

      7. mul-1-neg 83.4%

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

      8. remove-double-neg 83.4%

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

   7. Simplified 82.7%

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

   8. Taylor expanded in x around 0 82.6%

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

   9. Taylor expanded in z around 0 99.8%

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

       1. associate-*r/ 99.8%

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

       2. metadata-eval 99.8%

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

       3. +-commutative 99.8%

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

       4. mul-1-neg 99.8%

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

       5. associate-/l* 89.5%

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

       6. distribute-lft-neg-in 89.5%

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

       7. cancel-sign-sub 89.5%

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

       8. remove-double-neg 89.5%

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

   11. Simplified 89.5%

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

   if 3.7000000000000002 < x

   1. Initial program 89.9%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in x around inf 90.9%

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

      1. mul-1-neg 90.9%

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

      2. associate-/l* 98.6%

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

      3. distribute-rgt-neg-in 98.6%

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

      4. neg-sub0 98.6%

         \[\leadsto \left|x \cdot \color{blue}{\left(0 - \frac{z - 1}{y}\right)}\right| \]

      5. div0 98.6%

         \[\leadsto \left|x \cdot \left(\color{blue}{\frac{0}{y}} - \frac{z - 1}{y}\right)\right| \]

      6. div-sub 98.6%

         \[\leadsto \left|x \cdot \color{blue}{\frac{0 - \left(z - 1\right)}{y}}\right| \]

      7. associate-+l- 98.6%

         \[\leadsto \left|x \cdot \frac{\color{blue}{\left(0 - z\right) + 1}}{y}\right| \]

      8. neg-sub0 98.6%

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

      9. +-commutative 98.6%

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

      10. unsub-neg 98.6%

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

   7. Simplified 98.6%

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

      1. clear-num 98.5%

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

      2. un-div-inv 98.8%

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

   9. Applied egg-rr 98.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -15000000000:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 3.7:\\ \;\;\;\;\left|\frac{4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+110}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+20}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5e+110)
   (fabs (* (/ x y) (- 1.0 z)))
   (if (<= x 1.1e+20)
     (fabs (/ (- (+ x 4.0) (* x z)) y))
     (fabs (/ x (/ y (- 1.0 z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5e+110) {
		tmp = fabs(((x / y) * (1.0 - z)));
	} else if (x <= 1.1e+20) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((x / (y / (1.0 - z))));
	}
	return tmp;
}

Fortran

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+110)) then
        tmp = abs(((x / y) * (1.0d0 - z)))
    else if (x <= 1.1d+20) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else
        tmp = abs((x / (y / (1.0d0 - z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5e+110) {
		tmp = Math.abs(((x / y) * (1.0 - z)));
	} else if (x <= 1.1e+20) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((x / (y / (1.0 - z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5e+110:
		tmp = math.fabs(((x / y) * (1.0 - z)))
	elif x <= 1.1e+20:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((x / (y / (1.0 - z))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5e+110)
		tmp = abs(Float64(Float64(x / y) * Float64(1.0 - z)));
	elseif (x <= 1.1e+20)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(Float64(x / Float64(y / Float64(1.0 - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5e+110)
		tmp = abs(((x / y) * (1.0 - z)));
	elseif (x <= 1.1e+20)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((x / (y / (1.0 - z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5e+110], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.1e+20], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.99999999999999978e110

   1. Initial program 87.9%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in x around inf 82.9%

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

      1. mul-1-neg 82.9%

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

      2. *-commutative 82.9%

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

      3. associate-/l* 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. neg-sub0 99.9%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 99.9%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 99.9%

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

      8. +-commutative 99.9%

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

      9. 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 -4.99999999999999978e110 < x < 1.1e20

   1. Initial program 96.1%

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

      1. associate-*l/ 99.9%

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

      2. sub-div 99.9%

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

   4. Applied egg-rr 99.9%

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

   if 1.1e20 < x

   1. Initial program 89.2%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in x around inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. associate-/l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. neg-sub0 99.7%

         \[\leadsto \left|x \cdot \color{blue}{\left(0 - \frac{z - 1}{y}\right)}\right| \]

      5. div0 99.7%

         \[\leadsto \left|x \cdot \left(\color{blue}{\frac{0}{y}} - \frac{z - 1}{y}\right)\right| \]

      6. div-sub 99.7%

         \[\leadsto \left|x \cdot \color{blue}{\frac{0 - \left(z - 1\right)}{y}}\right| \]

      7. associate-+l- 99.7%

         \[\leadsto \left|x \cdot \frac{\color{blue}{\left(0 - z\right) + 1}}{y}\right| \]

      8. neg-sub0 99.7%

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

      9. +-commutative 99.7%

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

      10. unsub-neg 99.7%

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

   7. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+110}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+20}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-24} \lor \neg \left(x \leq 530000\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.5e-24) (not (<= x 530000.0)))
   (fabs (* x (/ (- 1.0 z) y)))
   (fabs (/ (- -4.0 x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.5e-24) || !(x <= 530000.0)) {
		tmp = fabs((x * ((1.0 - z) / y)));
	} else {
		tmp = fabs(((-4.0 - x) / y));
	}
	return tmp;
}

Fortran

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 <= (-8.5d-24)) .or. (.not. (x <= 530000.0d0))) then
        tmp = abs((x * ((1.0d0 - z) / y)))
    else
        tmp = abs((((-4.0d0) - x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.5e-24) || !(x <= 530000.0)) {
		tmp = Math.abs((x * ((1.0 - z) / y)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.5e-24) or not (x <= 530000.0):
		tmp = math.fabs((x * ((1.0 - z) / y)))
	else:
		tmp = math.fabs(((-4.0 - x) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.5e-24) || !(x <= 530000.0))
		tmp = abs(Float64(x * Float64(Float64(1.0 - z) / y)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.5e-24) || ~((x <= 530000.0)))
		tmp = abs((x * ((1.0 - z) / y)));
	else
		tmp = abs(((-4.0 - x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.5e-24], N[Not[LessEqual[x, 530000.0]], $MachinePrecision]], N[Abs[N[(x * N[(N[(1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.5000000000000002e-24 or 5.3e5 < x

   1. Initial program 90.2%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in x around inf 89.3%

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

      1. mul-1-neg 89.3%

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

      2. associate-/l* 98.8%

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

      3. distribute-rgt-neg-in 98.8%

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

      4. neg-sub0 98.8%

         \[\leadsto \left|x \cdot \color{blue}{\left(0 - \frac{z - 1}{y}\right)}\right| \]

      5. div0 98.8%

         \[\leadsto \left|x \cdot \left(\color{blue}{\frac{0}{y}} - \frac{z - 1}{y}\right)\right| \]

      6. div-sub 98.8%

         \[\leadsto \left|x \cdot \color{blue}{\frac{0 - \left(z - 1\right)}{y}}\right| \]

      7. associate-+l- 98.8%

         \[\leadsto \left|x \cdot \frac{\color{blue}{\left(0 - z\right) + 1}}{y}\right| \]

      8. neg-sub0 98.8%

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

      9. +-commutative 98.8%

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

      10. unsub-neg 98.8%

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

   7. Simplified 98.8%

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

   if -8.5000000000000002e-24 < x < 5.3e5

   1. Initial program 96.0%

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

      1. fabs-sub 96.0%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 89.2%

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

      4. fma-neg 89.2%

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

      5. distribute-neg-frac 89.2%

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

      6. +-commutative 89.2%

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

      7. distribute-neg-in 89.2%

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

      8. unsub-neg 89.2%

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

      9. metadata-eval 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in z around 0 78.6%

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

      1. associate-*r/ 78.6%

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

      2. distribute-lft-in 78.6%

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

      3. metadata-eval 78.6%

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

      4. neg-mul-1 78.6%

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

      5. sub-neg 78.6%

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

   7. Simplified 78.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-24} \lor \neg \left(x \leq 530000\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-28} \lor \neg \left(x \leq 41000\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.6e-28) (not (<= x 41000.0)))
   (fabs (* (/ x y) (- 1.0 z)))
   (fabs (/ (- -4.0 x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.6e-28) || !(x <= 41000.0)) {
		tmp = fabs(((x / y) * (1.0 - z)));
	} else {
		tmp = fabs(((-4.0 - x) / y));
	}
	return tmp;
}

Fortran

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 <= (-1.6d-28)) .or. (.not. (x <= 41000.0d0))) then
        tmp = abs(((x / y) * (1.0d0 - z)))
    else
        tmp = abs((((-4.0d0) - x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.6e-28) || !(x <= 41000.0)) {
		tmp = Math.abs(((x / y) * (1.0 - z)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.6e-28) or not (x <= 41000.0):
		tmp = math.fabs(((x / y) * (1.0 - z)))
	else:
		tmp = math.fabs(((-4.0 - x) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.6e-28) || !(x <= 41000.0))
		tmp = abs(Float64(Float64(x / y) * Float64(1.0 - z)));
	else
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.6e-28) || ~((x <= 41000.0)))
		tmp = abs(((x / y) * (1.0 - z)));
	else
		tmp = abs(((-4.0 - x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.6e-28], N[Not[LessEqual[x, 41000.0]], $MachinePrecision]], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.59999999999999991e-28 or 41000 < x

   1. Initial program 90.2%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in x around inf 89.3%

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

      1. mul-1-neg 89.3%

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

      2. *-commutative 89.3%

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

      3. associate-/l* 99.0%

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

      4. distribute-lft-neg-in 99.0%

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

      5. neg-sub0 99.0%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 99.0%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 99.0%

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

      8. +-commutative 99.0%

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

      9. unsub-neg 99.0%

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

   7. Simplified 99.0%

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

   if -1.59999999999999991e-28 < x < 41000

   1. Initial program 96.0%

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

      1. fabs-sub 96.0%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 89.2%

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

      4. fma-neg 89.2%

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

      5. distribute-neg-frac 89.2%

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

      6. +-commutative 89.2%

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

      7. distribute-neg-in 89.2%

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

      8. unsub-neg 89.2%

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

      9. metadata-eval 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in z around 0 78.6%

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

      1. associate-*r/ 78.6%

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

      2. distribute-lft-in 78.6%

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

      3. metadata-eval 78.6%

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

      4. neg-mul-1 78.6%

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

      5. sub-neg 78.6%

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

   7. Simplified 78.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-28} \lor \neg \left(x \leq 41000\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-24}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 2250:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.3e-24)
   (fabs (* (/ x y) (- 1.0 z)))
   (if (<= x 2250.0) (fabs (/ (- -4.0 x) y)) (fabs (/ x (/ y (- 1.0 z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e-24) {
		tmp = fabs(((x / y) * (1.0 - z)));
	} else if (x <= 2250.0) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((x / (y / (1.0 - z))));
	}
	return tmp;
}

Fortran

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 <= (-1.3d-24)) then
        tmp = abs(((x / y) * (1.0d0 - z)))
    else if (x <= 2250.0d0) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((x / (y / (1.0d0 - z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e-24) {
		tmp = Math.abs(((x / y) * (1.0 - z)));
	} else if (x <= 2250.0) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((x / (y / (1.0 - z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.3e-24:
		tmp = math.fabs(((x / y) * (1.0 - z)))
	elif x <= 2250.0:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((x / (y / (1.0 - z))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.3e-24)
		tmp = abs(Float64(Float64(x / y) * Float64(1.0 - z)));
	elseif (x <= 2250.0)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(x / Float64(y / Float64(1.0 - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.3e-24)
		tmp = abs(((x / y) * (1.0 - z)));
	elseif (x <= 2250.0)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((x / (y / (1.0 - z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.3e-24], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2250.0], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3e-24

   1. Initial program 90.6%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in x around inf 86.9%

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

      1. mul-1-neg 86.9%

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

      2. *-commutative 86.9%

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

      3. associate-/l* 98.3%

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

      4. distribute-lft-neg-in 98.3%

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

      5. neg-sub0 98.3%

         \[\leadsto \left|\color{blue}{\left(0 - \left(z - 1\right)\right)} \cdot \frac{x}{y}\right| \]

      6. associate-+l- 98.3%

         \[\leadsto \left|\color{blue}{\left(\left(0 - z\right) + 1\right)} \cdot \frac{x}{y}\right| \]

      7. neg-sub0 98.3%

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

      8. +-commutative 98.3%

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

      9. unsub-neg 98.3%

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

   7. Simplified 98.3%

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

   if -1.3e-24 < x < 2250

   1. Initial program 96.0%

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

      1. fabs-sub 96.0%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 89.2%

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

      4. fma-neg 89.2%

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

      5. distribute-neg-frac 89.2%

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

      6. +-commutative 89.2%

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

      7. distribute-neg-in 89.2%

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

      8. unsub-neg 89.2%

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

      9. metadata-eval 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in z around 0 78.6%

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

      1. associate-*r/ 78.6%

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

      2. distribute-lft-in 78.6%

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

      3. metadata-eval 78.6%

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

      4. neg-mul-1 78.6%

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

      5. sub-neg 78.6%

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

   7. Simplified 78.6%

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

   if 2250 < x

   1. Initial program 89.8%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in x around inf 91.8%

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

      1. mul-1-neg 91.8%

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

      2. associate-/l* 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

      4. neg-sub0 99.5%

         \[\leadsto \left|x \cdot \color{blue}{\left(0 - \frac{z - 1}{y}\right)}\right| \]

      5. div0 99.5%

         \[\leadsto \left|x \cdot \left(\color{blue}{\frac{0}{y}} - \frac{z - 1}{y}\right)\right| \]

      6. div-sub 99.5%

         \[\leadsto \left|x \cdot \color{blue}{\frac{0 - \left(z - 1\right)}{y}}\right| \]

      7. associate-+l- 99.5%

         \[\leadsto \left|x \cdot \frac{\color{blue}{\left(0 - z\right) + 1}}{y}\right| \]

      8. neg-sub0 99.5%

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

      9. +-commutative 99.5%

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

      10. unsub-neg 99.5%

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

   7. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.7%

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-24}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\\ \mathbf{elif}\;x \leq 2250:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-31)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (- (/ (+ x 4.0) y) (/ x (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-31) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs((((x + 4.0) / y) - (x / (y / z))));
	}
	return tmp;
}

Fortran

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 <= 5d-31) 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

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-31) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = Math.abs((((x + 4.0) / y) - (x / (y / z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5e-31:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	else:
		tmp = math.fabs((((x + 4.0) / y) - (x / (y / z))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e-31)
		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

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5e-31)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	else
		tmp = abs((((x + 4.0) / y) - (x / (y / z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e-31], 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 < 5e-31

   1. Initial program 91.4%

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

      1. associate-*l/ 88.6%

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

      2. sub-div 93.9%

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

   4. Applied egg-rr 93.9%

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

   if 5e-31 < y

   1. Initial program 96.3%

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

      1. associate-*l/ 95.8%

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

      2. associate-*r/ 99.9%

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

      3. clear-num 99.9%

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

      4. un-div-inv 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+24}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+50}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.2e+24)
   (fabs (/ x (/ y z)))
   (if (<= z 1.6e+50) (fabs (/ (- -4.0 x) y)) (fabs (* z (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.2e+24) {
		tmp = fabs((x / (y / z)));
	} else if (z <= 1.6e+50) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((z * (x / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.2d+24)) then
        tmp = abs((x / (y / z)))
    else if (z <= 1.6d+50) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((z * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.2e+24) {
		tmp = Math.abs((x / (y / z)));
	} else if (z <= 1.6e+50) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((z * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.2e+24:
		tmp = math.fabs((x / (y / z)))
	elif z <= 1.6e+50:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((z * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.2e+24)
		tmp = abs(Float64(x / Float64(y / z)));
	elseif (z <= 1.6e+50)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(z * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.2e+24)
		tmp = abs((x / (y / z)));
	elseif (z <= 1.6e+50)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((z * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.2e+24], N[Abs[N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.6e+50], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.2e24

   1. Initial program 94.3%

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

   3. Simplified 87.7%

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

   5. Taylor expanded in z around inf 68.1%

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

      1. mul-1-neg 68.1%

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

      2. distribute-frac-neg 68.1%

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

      3. distribute-lft-neg-in 68.1%

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

      4. *-commutative 68.1%

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

      5. associate-*r/ 75.2%

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

   7. Simplified 75.2%

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

      1. *-commutative 75.2%

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

      2. div-inv 75.1%

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

      3. associate-*l* 74.2%

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

      4. add-sqr-sqrt 40.9%

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

      5. sqrt-unprod 59.4%

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

      6. sqr-neg 59.4%

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

      7. sqrt-unprod 33.1%

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

      8. add-sqr-sqrt 74.2%

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

      9. associate-/r/ 74.2%

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

      10. div-inv 75.4%

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

   9. Applied egg-rr 75.4%

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

   if -1.2e24 < z < 1.59999999999999991e50

   1. Initial program 94.5%

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

      1. fabs-sub 94.5%

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

      2. associate-*l/ 97.2%

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

      3. associate-*r/ 97.2%

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

      4. fma-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. unsub-neg 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 98.1%

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

      1. associate-*r/ 98.1%

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

      2. distribute-lft-in 98.1%

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

      3. metadata-eval 98.1%

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

      4. neg-mul-1 98.1%

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

      5. sub-neg 98.1%

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

   7. Simplified 98.1%

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

   if 1.59999999999999991e50 < z

   1. Initial program 85.7%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in z around inf 70.9%

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

      1. mul-1-neg 70.9%

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

      2. distribute-frac-neg 70.9%

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

      3. distribute-lft-neg-in 70.9%

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

      4. *-commutative 70.9%

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

      5. associate-*r/ 82.5%

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

   7. Simplified 82.5%

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

      1. clear-num 82.5%

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

      2. un-div-inv 82.5%

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

      3. add-sqr-sqrt 45.1%

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

      4. sqrt-unprod 66.8%

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

      5. sqr-neg 66.8%

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

      6. sqrt-unprod 37.3%

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

      7. add-sqr-sqrt 82.5%

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

   9. Applied egg-rr 82.5%

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

       1. clear-num 82.5%

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

       2. associate-/r/ 82.5%

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

       3. clear-num 82.5%

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

   11. Applied egg-rr 82.5%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+24}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+50}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.3 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.3) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.3) || !(x <= 4.0)) {
		tmp = fabs((x / y));
	} else {
		tmp = fabs((4.0 / y));
	}
	return tmp;
}

Fortran

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 <= (-7.3d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs((x / y))
    else
        tmp = abs((4.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.3) || !(x <= 4.0)) {
		tmp = Math.abs((x / y));
	} else {
		tmp = Math.abs((4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -7.3) or not (x <= 4.0):
		tmp = math.fabs((x / y))
	else:
		tmp = math.fabs((4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7.3) || !(x <= 4.0))
		tmp = abs(Float64(x / y));
	else
		tmp = abs(Float64(4.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -7.3) || ~((x <= 4.0)))
		tmp = abs((x / y));
	else
		tmp = abs((4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7.3], 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 < -7.29999999999999982 or 4 < x

   1. Initial program 89.9%

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

   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 x around inf 89.3%

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

      1. mul-1-neg 89.3%

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

      2. associate-/l* 99.0%

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

      3. distribute-rgt-neg-in 99.0%

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

      4. neg-sub0 99.0%

         \[\leadsto \left|x \cdot \color{blue}{\left(0 - \frac{z - 1}{y}\right)}\right| \]

      5. div0 99.0%

         \[\leadsto \left|x \cdot \left(\color{blue}{\frac{0}{y}} - \frac{z - 1}{y}\right)\right| \]

      6. div-sub 99.0%

         \[\leadsto \left|x \cdot \color{blue}{\frac{0 - \left(z - 1\right)}{y}}\right| \]

      7. associate-+l- 99.0%

         \[\leadsto \left|x \cdot \frac{\color{blue}{\left(0 - z\right) + 1}}{y}\right| \]

      8. neg-sub0 99.0%

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

      9. +-commutative 99.0%

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

      10. unsub-neg 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in z around 0 63.8%

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

   if -7.29999999999999982 < x < 4

   1. Initial program 96.2%

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

   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 x around 0 75.9%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.3 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-26}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.22e-26)
   (fabs (* z (/ x y)))
   (if (<= x 4.0) (fabs (/ 4.0 y)) (fabs (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.22e-26) {
		tmp = fabs((z * (x / y)));
	} else if (x <= 4.0) {
		tmp = fabs((4.0 / y));
	} else {
		tmp = fabs((x / y));
	}
	return tmp;
}

Fortran

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 <= (-1.22d-26)) then
        tmp = abs((z * (x / y)))
    else if (x <= 4.0d0) then
        tmp = abs((4.0d0 / y))
    else
        tmp = abs((x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.22e-26) {
		tmp = Math.abs((z * (x / y)));
	} else if (x <= 4.0) {
		tmp = Math.abs((4.0 / y));
	} else {
		tmp = Math.abs((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.22e-26:
		tmp = math.fabs((z * (x / y)))
	elif x <= 4.0:
		tmp = math.fabs((4.0 / y))
	else:
		tmp = math.fabs((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.22e-26)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (x <= 4.0)
		tmp = abs(Float64(4.0 / y));
	else
		tmp = abs(Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.22e-26)
		tmp = abs((z * (x / y)));
	elseif (x <= 4.0)
		tmp = abs((4.0 / y));
	else
		tmp = abs((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.22e-26], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4.0], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.22e-26

   1. Initial program 90.6%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in z around inf 49.7%

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

      1. mul-1-neg 49.7%

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

      2. distribute-frac-neg 49.7%

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

      3. distribute-lft-neg-in 49.7%

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

      4. *-commutative 49.7%

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

      5. associate-*r/ 72.2%

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

   7. Simplified 72.2%

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

      1. clear-num 72.2%

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

      2. un-div-inv 68.5%

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

      3. add-sqr-sqrt 68.4%

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

      4. sqrt-unprod 53.9%

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

      5. sqr-neg 53.9%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 68.5%

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

   9. Applied egg-rr 68.5%

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

       1. clear-num 68.5%

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

       2. associate-/r/ 72.2%

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

       3. clear-num 72.2%

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

   11. Applied egg-rr 72.2%

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

   if -1.22e-26 < x < 4

   1. Initial program 96.0%

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

   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 x around 0 78.4%

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

   if 4 < x

   1. Initial program 89.9%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in x around inf 90.9%

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

      1. mul-1-neg 90.9%

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

      2. associate-/l* 98.6%

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

      3. distribute-rgt-neg-in 98.6%

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

      4. neg-sub0 98.6%

         \[\leadsto \left|x \cdot \color{blue}{\left(0 - \frac{z - 1}{y}\right)}\right| \]

      5. div0 98.6%

         \[\leadsto \left|x \cdot \left(\color{blue}{\frac{0}{y}} - \frac{z - 1}{y}\right)\right| \]

      6. div-sub 98.6%

         \[\leadsto \left|x \cdot \color{blue}{\frac{0 - \left(z - 1\right)}{y}}\right| \]

      7. associate-+l- 98.6%

         \[\leadsto \left|x \cdot \frac{\color{blue}{\left(0 - z\right) + 1}}{y}\right| \]

      8. neg-sub0 98.6%

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

      9. +-commutative 98.6%

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

      10. unsub-neg 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in z around 0 70.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-26}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 4:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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((4.0d0 / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 92.8%

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

3. Simplified 94.3%

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

5. Taylor expanded in x around 0 37.2%

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

6. Final simplification 37.2%

   \[\leadsto \left|\frac{4}{y}\right| \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))