fabs fraction 1

Percentage Accurate: 91.9% → 99.2%

Time: 8.3s

Alternatives: 11

Speedup: 0.9×

• 20.3% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{elif}\;t\_0 \leq 1.1 \cdot 10^{+296}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (- (/ (+ x 4.0) y) (* z (/ x y))))))
   (if (<= t_0 5e-27)
     (fabs (/ (- (+ x 4.0) (* x z)) y))
     (if (<= t_0 1.1e+296) t_0 (fabs (* (/ x y) (+ z -1.0)))))))

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 <= 5e-27) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else if (t_0 <= 1.1e+296) {
		tmp = t_0;
	} else {
		tmp = fabs(((x / y) * (z + -1.0)));
	}
	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 <= 5d-27) then
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    else if (t_0 <= 1.1d+296) then
        tmp = t_0
    else
        tmp = abs(((x / y) * (z + (-1.0d0))))
    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 <= 5e-27) {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	} else if (t_0 <= 1.1e+296) {
		tmp = t_0;
	} else {
		tmp = Math.abs(((x / y) * (z + -1.0)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((((x + 4.0) / y) - (z * (x / y))))
	tmp = 0
	if t_0 <= 5e-27:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	elif t_0 <= 1.1e+296:
		tmp = t_0
	else:
		tmp = math.fabs(((x / y) * (z + -1.0)))
	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 <= 5e-27)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	elseif (t_0 <= 1.1e+296)
		tmp = t_0;
	else
		tmp = abs(Float64(Float64(x / y) * Float64(z + -1.0)));
	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 <= 5e-27)
		tmp = abs((((x + 4.0) - (x * z)) / y));
	elseif (t_0 <= 1.1e+296)
		tmp = t_0;
	else
		tmp = abs(((x / y) * (z + -1.0)));
	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[LessEqual[t$95$0, 5e-27], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 1.1e+296], t$95$0, N[Abs[N[(N[(x / y), $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 5.0000000000000002e-27

   1. Initial program 91.6%

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
   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 5.0000000000000002e-27 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z))) < 1.10000000000000007e296

   1. Initial program 99.8%

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

   if 1.10000000000000007e296 < (fabs.f64 (-.f64 (/.f64 (+.f64 x #s(literal 4 binary64)) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 81.8%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in x around inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-/l* 100.0%

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

      4. distribute-lft-neg-in 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate-+l- 100.0%

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

      7. neg-sub0 100.0%

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

      8. +-commutative 100.0%

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

      9. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{elif}\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right| \leq 1.1 \cdot 10^{+296}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+52}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.5e+52)
   (fabs (/ (- (+ x 4.0) (* x z)) y))
   (fabs (fma x (/ z y) (/ (- -4.0 x) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.5e+52) {
		tmp = fabs((((x + 4.0) - (x * z)) / y));
	} else {
		tmp = fabs(fma(x, (z / y), ((-4.0 - x) / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.5e+52)
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	else
		tmp = abs(fma(x, Float64(z / y), Float64(Float64(-4.0 - x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.5e+52], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision] + N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.49999999999999994e52

   1. Initial program 93.9%

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

      1. associate-*l/ 94.9%

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

      2. sub-div 97.3%

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

   4. Applied egg-rr 97.3%

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

   if 9.49999999999999994e52 < y

   1. Initial program 93.8%

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

      1. fabs-sub 93.8%

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

      2. associate-*l/ 85.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-neg 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. +-commutative 99.7%

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

      7. distribute-neg-in 99.7%

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

      8. unsub-neg 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+52}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x, \frac{z}{y}, \frac{-4 - x}{y}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|z \cdot \frac{x}{y}\right|\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-14}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+136} \lor \neg \left(x \leq 1.15 \cdot 10^{+234}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (* z (/ x y)))))
   (if (<= x -6.5e-26)
     t_0
     (if (<= x 6.2e-14)
       (fabs (/ 4.0 y))
       (if (or (<= x 7.5e+136) (not (<= x 1.15e+234))) t_0 (fabs (/ x y)))))))

C

double code(double x, double y, double z) {
	double t_0 = fabs((z * (x / y)));
	double tmp;
	if (x <= -6.5e-26) {
		tmp = t_0;
	} else if (x <= 6.2e-14) {
		tmp = fabs((4.0 / y));
	} else if ((x <= 7.5e+136) || !(x <= 1.15e+234)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = abs((z * (x / y)))
    if (x <= (-6.5d-26)) then
        tmp = t_0
    else if (x <= 6.2d-14) then
        tmp = abs((4.0d0 / y))
    else if ((x <= 7.5d+136) .or. (.not. (x <= 1.15d+234))) then
        tmp = t_0
    else
        tmp = abs((x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.abs((z * (x / y)));
	double tmp;
	if (x <= -6.5e-26) {
		tmp = t_0;
	} else if (x <= 6.2e-14) {
		tmp = Math.abs((4.0 / y));
	} else if ((x <= 7.5e+136) || !(x <= 1.15e+234)) {
		tmp = t_0;
	} else {
		tmp = Math.abs((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.fabs((z * (x / y)))
	tmp = 0
	if x <= -6.5e-26:
		tmp = t_0
	elif x <= 6.2e-14:
		tmp = math.fabs((4.0 / y))
	elif (x <= 7.5e+136) or not (x <= 1.15e+234):
		tmp = t_0
	else:
		tmp = math.fabs((x / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = abs(Float64(z * Float64(x / y)))
	tmp = 0.0
	if (x <= -6.5e-26)
		tmp = t_0;
	elseif (x <= 6.2e-14)
		tmp = abs(Float64(4.0 / y));
	elseif ((x <= 7.5e+136) || !(x <= 1.15e+234))
		tmp = t_0;
	else
		tmp = abs(Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = abs((z * (x / y)));
	tmp = 0.0;
	if (x <= -6.5e-26)
		tmp = t_0;
	elseif (x <= 6.2e-14)
		tmp = abs((4.0 / y));
	elseif ((x <= 7.5e+136) || ~((x <= 1.15e+234)))
		tmp = t_0;
	else
		tmp = abs((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -6.5e-26], t$95$0, If[LessEqual[x, 6.2e-14], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 7.5e+136], N[Not[LessEqual[x, 1.15e+234]], $MachinePrecision]], t$95$0, N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5e-26 or 6.20000000000000009e-14 < x < 7.5000000000000002e136 or 1.15e234 < x

   1. Initial program 93.2%

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

      1. fabs-sub 93.2%

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

      2. associate-*l/ 87.3%

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

      3. associate-*r/ 96.4%

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

      4. fma-neg 98.1%

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

      5. distribute-neg-frac 98.1%

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

      6. +-commutative 98.1%

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

      7. distribute-neg-in 98.1%

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

      8. unsub-neg 98.1%

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

      9. metadata-eval 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in z around inf 84.8%

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

      1. +-commutative 84.8%

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

      2. associate-*r/ 84.8%

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

      3. distribute-lft-in 84.8%

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

      4. metadata-eval 84.8%

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

      5. neg-mul-1 84.8%

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

      6. sub-neg 84.8%

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

      7. associate-/r* 83.2%

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

   7. Simplified 83.2%

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

   8. Taylor expanded in z around inf 53.5%

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

      1. *-commutative 53.5%

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

      2. associate-*r/ 69.8%

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

   10. Simplified 69.8%

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

   if -6.5e-26 < x < 6.20000000000000009e-14

   1. Initial program 95.4%

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

   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 79.8%

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

   if 7.5000000000000002e136 < x < 1.15e234

   1. Initial program 86.7%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in x around inf 93.5%

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

      1. mul-1-neg 93.5%

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

      2. *-commutative 93.5%

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

      3. associate-/l* 100.0%

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

      4. distribute-lft-neg-in 100.0%

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

      5. neg-sub0 100.0%

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

      6. associate-+l- 100.0%

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

      7. neg-sub0 100.0%

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

      8. +-commutative 100.0%

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

      9. 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 81.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-26}:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-14}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+136} \lor \neg \left(x \leq 1.15 \cdot 10^{+234}\right):\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -200 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot \left(\frac{4}{z} - x\right)}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -200.0) (not (<= x 4.0)))
   (fabs (* (/ x y) (+ z -1.0)))
   (fabs (/ (* z (- (/ 4.0 z) x)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -200.0) || !(x <= 4.0)) {
		tmp = fabs(((x / y) * (z + -1.0)));
	} else {
		tmp = fabs(((z * ((4.0 / 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 ((x <= (-200.0d0)) .or. (.not. (x <= 4.0d0))) then
        tmp = abs(((x / y) * (z + (-1.0d0))))
    else
        tmp = abs(((z * ((4.0d0 / z) - x)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -200.0) || !(x <= 4.0)) {
		tmp = Math.abs(((x / y) * (z + -1.0)));
	} else {
		tmp = Math.abs(((z * ((4.0 / z) - x)) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -200.0) or not (x <= 4.0):
		tmp = math.fabs(((x / y) * (z + -1.0)))
	else:
		tmp = math.fabs(((z * ((4.0 / z) - x)) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -200.0) || !(x <= 4.0))
		tmp = abs(Float64(Float64(x / y) * Float64(z + -1.0)));
	else
		tmp = abs(Float64(Float64(z * Float64(Float64(4.0 / z) - x)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -200.0) || ~((x <= 4.0)))
		tmp = abs(((x / y) * (z + -1.0)));
	else
		tmp = abs(((z * ((4.0 / z) - x)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -200.0], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(z * N[(N[(4.0 / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -200 or 4 < x

   1. Initial program 91.8%

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

   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 88.3%

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

      1. mul-1-neg 88.3%

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

      2. *-commutative 88.3%

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

      3. associate-/l* 98.0%

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

      4. distribute-lft-neg-in 98.0%

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

      5. neg-sub0 98.0%

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

      6. associate-+l- 98.0%

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

      7. neg-sub0 98.0%

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

      8. +-commutative 98.0%

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

      9. unsub-neg 98.0%

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

   7. Simplified 98.0%

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

   if -200 < x < 4

   1. Initial program 95.8%

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

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

   5. Taylor expanded in z around -inf 99.8%

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

      1. associate-*r* 99.8%

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

      2. neg-mul-1 99.8%

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

      3. distribute-lft-out-- 99.8%

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

      4. associate-*r* 99.8%

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

      5. distribute-lft-neg-in 99.8%

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

      6. *-commutative 99.8%

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

      7. neg-mul-1 99.8%

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

      8. remove-double-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 98.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z \cdot \left(\frac{4}{z} - x\right)}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+70} \lor \neg \left(x \leq 9 \cdot 10^{+51}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2e+70) (not (<= x 9e+51)))
   (fabs (* (/ x y) (+ z -1.0)))
   (fabs (/ (- (+ x 4.0) (* x z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e+70) || !(x <= 9e+51)) {
		tmp = fabs(((x / y) * (z + -1.0)));
	} else {
		tmp = fabs((((x + 4.0) - (x * z)) / 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 <= (-2d+70)) .or. (.not. (x <= 9d+51))) then
        tmp = abs(((x / y) * (z + (-1.0d0))))
    else
        tmp = abs((((x + 4.0d0) - (x * z)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e+70) || !(x <= 9e+51)) {
		tmp = Math.abs(((x / y) * (z + -1.0)));
	} else {
		tmp = Math.abs((((x + 4.0) - (x * z)) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2e+70) or not (x <= 9e+51):
		tmp = math.fabs(((x / y) * (z + -1.0)))
	else:
		tmp = math.fabs((((x + 4.0) - (x * z)) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2e+70) || !(x <= 9e+51))
		tmp = abs(Float64(Float64(x / y) * Float64(z + -1.0)));
	else
		tmp = abs(Float64(Float64(Float64(x + 4.0) - Float64(x * z)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2e+70) || ~((x <= 9e+51)))
		tmp = abs(((x / y) * (z + -1.0)));
	else
		tmp = abs((((x + 4.0) - (x * z)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2e+70], N[Not[LessEqual[x, 9e+51]], $MachinePrecision]], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] - N[(x * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000015e70 or 8.9999999999999999e51 < x

   1. Initial program 90.5%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in x around inf 87.3%

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

      1. mul-1-neg 87.3%

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

      2. *-commutative 87.3%

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

      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 -2.00000000000000015e70 < x < 8.9999999999999999e51

   1. Initial program 95.8%

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

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+70} \lor \neg \left(x \leq 9 \cdot 10^{+51}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-26} \lor \neg \left(x \leq 3000000000\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.6e-26) (not (<= x 3000000000.0)))
   (fabs (* x (/ (- 1.0 z) y)))
   (fabs (/ (- -4.0 x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.6e-26) || !(x <= 3000000000.0)) {
		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 <= (-2.6d-26)) .or. (.not. (x <= 3000000000.0d0))) 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 <= -2.6e-26) || !(x <= 3000000000.0)) {
		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 <= -2.6e-26) or not (x <= 3000000000.0):
		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 <= -2.6e-26) || !(x <= 3000000000.0))
		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 <= -2.6e-26) || ~((x <= 3000000000.0)))
		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, -2.6e-26], N[Not[LessEqual[x, 3000000000.0]], $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 < -2.6000000000000001e-26 or 3e9 < x

   1. Initial program 92.1%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in x around inf 89.1%

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

      1. *-commutative 89.1%

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

      2. associate-/l* 98.3%

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

      3. associate-*r* 98.3%

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

      4. *-commutative 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. mul-1-neg 98.3%

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

      7. neg-sub0 98.3%

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

      8. associate-+l- 98.3%

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

      9. neg-sub0 98.3%

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

      10. +-commutative 98.3%

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

      11. 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 -2.6000000000000001e-26 < x < 3e9

   1. Initial program 95.6%

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

   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 z around 0 79.5%

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

      1. +-commutative 79.5%

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

      2. rem-square-sqrt 29.5%

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

      3. fabs-sqr 29.5%

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

      4. rem-square-sqrt 79.5%

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

      5. fabs-neg 79.5%

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

      6. distribute-neg-frac 79.5%

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

      7. distribute-neg-in 79.5%

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

      8. metadata-eval 79.5%

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

      9. +-commutative 79.5%

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

      10. sub-neg 79.5%

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

      11. rem-square-sqrt 49.5%

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

      12. fabs-sqr 49.5%

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

      13. rem-square-sqrt 79.5%

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

   7. Simplified 79.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-26} \lor \neg \left(x \leq 3000000000\right):\\ \;\;\;\;\left|x \cdot \frac{1 - z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-29} \lor \neg \left(x \leq 2200000000\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.5e-29) (not (<= x 2200000000.0)))
   (fabs (* (/ x y) (+ z -1.0)))
   (fabs (/ (- -4.0 x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.5e-29) || !(x <= 2200000000.0)) {
		tmp = fabs(((x / y) * (z + -1.0)));
	} 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 <= (-2.5d-29)) .or. (.not. (x <= 2200000000.0d0))) then
        tmp = abs(((x / y) * (z + (-1.0d0))))
    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 <= -2.5e-29) || !(x <= 2200000000.0)) {
		tmp = Math.abs(((x / y) * (z + -1.0)));
	} else {
		tmp = Math.abs(((-4.0 - x) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.5e-29) or not (x <= 2200000000.0):
		tmp = math.fabs(((x / y) * (z + -1.0)))
	else:
		tmp = math.fabs(((-4.0 - x) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.5e-29) || !(x <= 2200000000.0))
		tmp = abs(Float64(Float64(x / y) * Float64(z + -1.0)));
	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 <= -2.5e-29) || ~((x <= 2200000000.0)))
		tmp = abs(((x / y) * (z + -1.0)));
	else
		tmp = abs(((-4.0 - x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.5e-29], N[Not[LessEqual[x, 2200000000.0]], $MachinePrecision]], N[Abs[N[(N[(x / y), $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.49999999999999993e-29 or 2.2e9 < x

   1. Initial program 92.1%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in x around 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. *-commutative 89.1%

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

      3. associate-/l* 98.5%

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

      4. distribute-lft-neg-in 98.5%

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

      5. neg-sub0 98.5%

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

      6. associate-+l- 98.5%

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

      7. neg-sub0 98.5%

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

      8. +-commutative 98.5%

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

      9. unsub-neg 98.5%

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

   7. Simplified 98.5%

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

   if -2.49999999999999993e-29 < x < 2.2e9

   1. Initial program 95.6%

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

   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 z around 0 79.5%

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

      1. +-commutative 79.5%

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

      2. rem-square-sqrt 29.5%

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

      3. fabs-sqr 29.5%

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

      4. rem-square-sqrt 79.5%

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

      5. fabs-neg 79.5%

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

      6. distribute-neg-frac 79.5%

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

      7. distribute-neg-in 79.5%

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

      8. metadata-eval 79.5%

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

      9. +-commutative 79.5%

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

      10. sub-neg 79.5%

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

      11. rem-square-sqrt 49.5%

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

      12. fabs-sqr 49.5%

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

      13. rem-square-sqrt 79.5%

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

   7. Simplified 79.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-29} \lor \neg \left(x \leq 2200000000\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(z + -1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -210000000000:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+75}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -210000000000.0)
   (fabs (* z (/ x y)))
   (if (<= z 1.45e+75) (fabs (/ (- -4.0 x) y)) (fabs (/ z (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -210000000000.0) {
		tmp = fabs((z * (x / y)));
	} else if (z <= 1.45e+75) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((z / (y / x)));
	}
	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 <= (-210000000000.0d0)) then
        tmp = abs((z * (x / y)))
    else if (z <= 1.45d+75) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((z / (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -210000000000.0) {
		tmp = Math.abs((z * (x / y)));
	} else if (z <= 1.45e+75) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((z / (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -210000000000.0:
		tmp = math.fabs((z * (x / y)))
	elif z <= 1.45e+75:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((z / (y / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -210000000000.0)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (z <= 1.45e+75)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(z / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -210000000000.0)
		tmp = abs((z * (x / y)));
	elseif (z <= 1.45e+75)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((z / (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -210000000000.0], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 1.45e+75], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1e11

   1. Initial program 98.1%

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

      1. fabs-sub 98.1%

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

      2. associate-*l/ 89.4%

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

      3. associate-*r/ 90.9%

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

      4. fma-neg 90.9%

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

      5. distribute-neg-frac 90.9%

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

      6. +-commutative 90.9%

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

      7. distribute-neg-in 90.9%

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

      8. unsub-neg 90.9%

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

      9. metadata-eval 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in z around inf 94.5%

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

      1. +-commutative 94.5%

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

      2. associate-*r/ 94.5%

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

      3. distribute-lft-in 94.5%

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

      4. metadata-eval 94.5%

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

      5. neg-mul-1 94.5%

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

      6. sub-neg 94.5%

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

      7. associate-/r* 94.6%

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

   7. Simplified 94.6%

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

   8. Taylor expanded in z around inf 68.0%

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

      1. *-commutative 68.0%

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

      2. associate-*r/ 76.6%

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

   10. Simplified 76.6%

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

   if -2.1e11 < z < 1.4499999999999999e75

   1. Initial program 94.6%

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

   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 95.1%

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

      1. +-commutative 95.1%

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

      2. rem-square-sqrt 43.2%

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

      3. fabs-sqr 43.2%

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

      4. rem-square-sqrt 95.1%

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

      5. fabs-neg 95.1%

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

      6. distribute-neg-frac 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. metadata-eval 95.1%

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

      9. +-commutative 95.1%

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

      10. sub-neg 95.1%

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

      11. rem-square-sqrt 51.3%

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

      12. fabs-sqr 51.3%

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

      13. rem-square-sqrt 95.1%

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

   7. Simplified 95.1%

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

   if 1.4499999999999999e75 < z

   1. Initial program 87.8%

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

   3. Simplified 87.8%

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

   5. Taylor expanded in z around inf 69.8%

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

      1. associate-*r/ 69.8%

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

      2. neg-mul-1 69.8%

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

      3. distribute-lft-neg-in 69.8%

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

      4. *-commutative 69.8%

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

      5. associate-*r/ 74.3%

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

   7. Simplified 74.3%

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

      1. add-sqr-sqrt 45.9%

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

      2. sqrt-unprod 61.9%

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

      3. sqr-neg 61.9%

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

      4. sqrt-unprod 28.1%

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

      5. add-sqr-sqrt 74.3%

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

      6. clear-num 74.2%

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

      7. un-div-inv 74.3%

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

   9. Applied egg-rr 74.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -210000000000:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+75}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -215000000000:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+75}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -215000000000.0)
   (fabs (* z (/ x y)))
   (if (<= z 3e+75) (fabs (/ (- -4.0 x) y)) (fabs (* x (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -215000000000.0) {
		tmp = fabs((z * (x / y)));
	} else if (z <= 3e+75) {
		tmp = fabs(((-4.0 - x) / y));
	} else {
		tmp = fabs((x * (z / 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 <= (-215000000000.0d0)) then
        tmp = abs((z * (x / y)))
    else if (z <= 3d+75) then
        tmp = abs((((-4.0d0) - x) / y))
    else
        tmp = abs((x * (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -215000000000.0) {
		tmp = Math.abs((z * (x / y)));
	} else if (z <= 3e+75) {
		tmp = Math.abs(((-4.0 - x) / y));
	} else {
		tmp = Math.abs((x * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -215000000000.0:
		tmp = math.fabs((z * (x / y)))
	elif z <= 3e+75:
		tmp = math.fabs(((-4.0 - x) / y))
	else:
		tmp = math.fabs((x * (z / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -215000000000.0)
		tmp = abs(Float64(z * Float64(x / y)));
	elseif (z <= 3e+75)
		tmp = abs(Float64(Float64(-4.0 - x) / y));
	else
		tmp = abs(Float64(x * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -215000000000.0)
		tmp = abs((z * (x / y)));
	elseif (z <= 3e+75)
		tmp = abs(((-4.0 - x) / y));
	else
		tmp = abs((x * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -215000000000.0], N[Abs[N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 3e+75], N[Abs[N[(N[(-4.0 - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.15e11

   1. Initial program 98.1%

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

      1. fabs-sub 98.1%

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

      2. associate-*l/ 89.4%

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

      3. associate-*r/ 90.9%

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

      4. fma-neg 90.9%

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

      5. distribute-neg-frac 90.9%

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

      6. +-commutative 90.9%

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

      7. distribute-neg-in 90.9%

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

      8. unsub-neg 90.9%

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

      9. metadata-eval 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in z around inf 94.5%

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

      1. +-commutative 94.5%

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

      2. associate-*r/ 94.5%

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

      3. distribute-lft-in 94.5%

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

      4. metadata-eval 94.5%

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

      5. neg-mul-1 94.5%

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

      6. sub-neg 94.5%

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

      7. associate-/r* 94.6%

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

   7. Simplified 94.6%

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

   8. Taylor expanded in z around inf 68.0%

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

      1. *-commutative 68.0%

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

      2. associate-*r/ 76.6%

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

   10. Simplified 76.6%

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

   if -2.15e11 < z < 3e75

   1. Initial program 94.6%

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

   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 95.1%

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

      1. +-commutative 95.1%

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

      2. rem-square-sqrt 43.2%

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

      3. fabs-sqr 43.2%

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

      4. rem-square-sqrt 95.1%

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

      5. fabs-neg 95.1%

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

      6. distribute-neg-frac 95.1%

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

      7. distribute-neg-in 95.1%

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

      8. metadata-eval 95.1%

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

      9. +-commutative 95.1%

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

      10. sub-neg 95.1%

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

      11. rem-square-sqrt 51.3%

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

      12. fabs-sqr 51.3%

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

      13. rem-square-sqrt 95.1%

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

   7. Simplified 95.1%

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

   if 3e75 < z

   1. Initial program 87.8%

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

   3. Simplified 87.8%

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

   5. Taylor expanded in z around inf 69.8%

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

      1. mul-1-neg 69.8%

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

      2. distribute-frac-neg2 69.8%

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

      3. associate-/l* 75.0%

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

   7. Simplified 75.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -215000000000:\\ \;\;\;\;\left|z \cdot \frac{x}{y}\right|\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+75}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -10.5) (not (<= x 4.0))) (fabs (/ x y)) (fabs (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -10.5) || !(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 <= (-10.5d0)) .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 <= -10.5) || !(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 <= -10.5) 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 <= -10.5) || !(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 <= -10.5) || ~((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, -10.5], N[Not[LessEqual[x, 4.0]], $MachinePrecision]], N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision], N[Abs[N[(4.0 / y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -10.5 or 4 < x

   1. Initial program 91.9%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in x around inf 88.4%

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

      1. mul-1-neg 88.4%

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

      2. *-commutative 88.4%

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

      3. associate-/l* 98.0%

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

      4. distribute-lft-neg-in 98.0%

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

      5. neg-sub0 98.0%

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

      6. associate-+l- 98.0%

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

      7. neg-sub0 98.0%

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

      8. +-commutative 98.0%

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

      9. 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 56.0%

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

   if -10.5 < x < 4

   1. Initial program 95.7%

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

   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 75.1%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10.5 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.2% 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 93.8%

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

3. Simplified 95.1%

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

5. Taylor expanded in x around 0 40.9%

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

6. Final simplification 40.9%

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

Reproduce

herbie shell --seed 2024071 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))