fabs fraction 1

Average Error: 1.6 → 0.5

Time: 9.2s

Precision: binary64

Cost: 7368

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ \mathbf{if}\;x \leq -1 \cdot 10^{+46}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{elif}\;x \leq 2.260811174163227 \cdot 10^{-155}:\\ \;\;\;\;\left|t_0 - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_0 - z \cdot \frac{x}{y}\right|\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x 4.0) y)))
   (if (<= x -1e+46)
     (fabs (/ x (/ y (- 1.0 z))))
     (if (<= x 2.260811174163227e-155)
       (fabs (- t_0 (/ (* x z) y)))
       (fabs (- t_0 (* z (/ x y))))))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = (x + 4.0) / y;
	double tmp;
	if (x <= -1e+46) {
		tmp = fabs((x / (y / (1.0 - z))));
	} else if (x <= 2.260811174163227e-155) {
		tmp = fabs((t_0 - ((x * z) / y)));
	} else {
		tmp = fabs((t_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
    code = abs((((x + 4.0d0) / y) - ((x / y) * z)))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + 4.0d0) / y
    if (x <= (-1d+46)) then
        tmp = abs((x / (y / (1.0d0 - z))))
    else if (x <= 2.260811174163227d-155) then
        tmp = abs((t_0 - ((x * z) / y)))
    else
        tmp = abs((t_0 - (z * (x / y))))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double t_0 = (x + 4.0) / y;
	double tmp;
	if (x <= -1e+46) {
		tmp = Math.abs((x / (y / (1.0 - z))));
	} else if (x <= 2.260811174163227e-155) {
		tmp = Math.abs((t_0 - ((x * z) / y)));
	} else {
		tmp = Math.abs((t_0 - (z * (x / y))));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x + 4.0) / y
	tmp = 0
	if x <= -1e+46:
		tmp = math.fabs((x / (y / (1.0 - z))))
	elif x <= 2.260811174163227e-155:
		tmp = math.fabs((t_0 - ((x * z) / y)))
	else:
		tmp = math.fabs((t_0 - (z * (x / y))))
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(Float64(x + 4.0) / y)
	tmp = 0.0
	if (x <= -1e+46)
		tmp = abs(Float64(x / Float64(y / Float64(1.0 - z))));
	elseif (x <= 2.260811174163227e-155)
		tmp = abs(Float64(t_0 - Float64(Float64(x * z) / y)));
	else
		tmp = abs(Float64(t_0 - Float64(z * Float64(x / y))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = (x + 4.0) / y;
	tmp = 0.0;
	if (x <= -1e+46)
		tmp = abs((x / (y / (1.0 - z))));
	elseif (x <= 2.260811174163227e-155)
		tmp = abs((t_0 - ((x * z) / y)));
	else
		tmp = abs((t_0 - (z * (x / y))));
	end
	tmp_2 = tmp;
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]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -1e+46], N[Abs[N[(x / N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.260811174163227e-155], N[Abs[N[(t$95$0 - N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$0 - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.9999999999999999e45

   1. Initial program 0.1

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

   2. Taylor expanded in x around inf 0.3

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

   3. Simplified 0.1

      \[\leadsto \left|\color{blue}{\frac{x}{\frac{y}{1 - z}}}\right| \]
      Proof
      (/.f64 x (/.f64 y (-.f64 1 z))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 y (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 z))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 y (+.f64 1 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 z))))): 0 points increase in error, 0 points decrease in error
      (/.f64 x (/.f64 y (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 z) 1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 x y) (+.f64 (*.f64 -1 z) 1))): 27 points increase in error, 26 points decrease in error
      (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 (*.f64 -1 z) 1) (/.f64 x y))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (+.f64 (*.f64 -1 z) 1) x) y)): 47 points increase in error, 17 points decrease in error
      (Rewrite=> associate-/l*_binary64 (/.f64 (+.f64 (*.f64 -1 z) 1) (/.f64 y x))): 41 points increase in error, 60 points decrease in error
      (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 1 (*.f64 -1 z))) (/.f64 y x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (Rewrite<= neg-mul-1_binary64 (neg.f64 z))) (/.f64 y x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> unsub-neg_binary64 (-.f64 1 z)) (/.f64 y x)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 1 (/.f64 y x)) (/.f64 z (/.f64 y x)))): 1 points increase in error, 1 points decrease in error
      (-.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 y) x)) (/.f64 z (/.f64 y x))): 27 points increase in error, 25 points decrease in error
      (-.f64 (*.f64 (/.f64 1 y) x) (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 z y) x))): 30 points increase in error, 19 points decrease in error
      (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x (-.f64 (/.f64 1 y) (/.f64 z y)))): 3 points increase in error, 1 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (/.f64 1 y) (/.f64 z y)) x)): 0 points increase in error, 0 points decrease in error

   if -9.9999999999999999e45 < x < 2.2608111741632268e-155

   1. Initial program 2.3

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

   2. Taylor expanded in x around 0 0.2

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

   if 2.2608111741632268e-155 < x

   1. Initial program 1.2

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+46}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{elif}\;x \leq 2.260811174163227 \cdot 10^{-155}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	7368

\[\begin{array}{l} t_0 := \left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{if}\;x \leq -1 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{+100}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	4.8
Cost	7244

\[\begin{array}{l} t_0 := \left|\frac{4 - x \cdot z}{y}\right|\\ \mathbf{if}\;z \leq -30786486.299687263:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 9.43084176186724 \cdot 10^{-9}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{elif}\;z \leq 10^{+175}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \end{array} \]

Alternative 3
Error	1.1
Cost	7240

\[\begin{array}{l} t_0 := \left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{if}\;x \leq -591353427706.8567:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.9908120938300156 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.3
Cost	7240

\[\begin{array}{l} t_0 := \left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{if}\;x \leq -1 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{+100}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.0
Cost	7112

\[\begin{array}{l} t_0 := \left|\frac{x}{\frac{y}{1 - z}}\right|\\ \mathbf{if}\;x \leq -591353427706.8567:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.9908120938300156 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{4 - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	19.3
Cost	6988

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.369526861616593:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.375575257912459 \cdot 10^{-63}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 1.4475894409323352 \cdot 10^{-8}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	19.2
Cost	6988

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.369526861616593:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.375575257912459 \cdot 10^{-63}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;x \leq 1.4475894409323352 \cdot 10^{-8}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	11.6
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -9.1 \cdot 10^{+77}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+61}:\\ \;\;\;\;\left|\frac{x + 4}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \end{array} \]

Alternative 9
Error	18.9
Cost	6856

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -591353427706.8567:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4475894409323352 \cdot 10^{-8}:\\ \;\;\;\;\frac{4}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	46.7
Cost	6592

\[\left|\frac{x}{y}\right| \]

Reproduce

herbie shell --seed 2022308 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))