fabs fraction 1

Average Accuracy: 91.6% → 99.4%

Time: 11.7s

Precision: binary64

Cost: 21576

• 20.4% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-19}:\\ \;\;\;\;\left|\frac{x - \mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)\right|\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z)))))
   (if (<= t_0 4e-19)
     (fabs (/ (- x (fma x z -4.0)) y))
     (if (<= t_0 5e+291)
       t_0
       (fabs (+ (/ 4.0 y) (* x (- (/ 1.0 y) (/ z 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 = fabs((((x + 4.0) / y) - ((x / y) * z)));
	double tmp;
	if (t_0 <= 4e-19) {
		tmp = fabs(((x - fma(x, z, -4.0)) / y));
	} else if (t_0 <= 5e+291) {
		tmp = t_0;
	} else {
		tmp = fabs(((4.0 / y) + (x * ((1.0 / y) - (z / 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 = abs(Float64(Float64(Float64(x + 4.0) / y) - Float64(Float64(x / y) * z)))
	tmp = 0.0
	if (t_0 <= 4e-19)
		tmp = abs(Float64(Float64(x - fma(x, z, -4.0)) / y));
	elseif (t_0 <= 5e+291)
		tmp = t_0;
	else
		tmp = abs(Float64(Float64(4.0 / y) + Float64(x * Float64(Float64(1.0 / y) - Float64(z / y)))));
	end
	return 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[Abs[N[(N[(N[(x + 4.0), $MachinePrecision] / y), $MachinePrecision] - N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 4e-19], N[Abs[N[(N[(x - N[(x * z + -4.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 5e+291], t$95$0, N[Abs[N[(N[(4.0 / y), $MachinePrecision] + N[(x * N[(N[(1.0 / y), $MachinePrecision] - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))) < 3.9999999999999999e-19

   1. Initial program 91.2%

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

   2. Simplified 100.0%

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

      [Start] 91.2	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      fabs-neg [<=] 91.2	\[ \color{blue}{\left|-\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right|} \]
      sub-neg [=>] 91.2	\[ \left|-\color{blue}{\left(\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)\right)}\right| \]
      distribute-neg-in [=>] 91.2	\[ \left|\color{blue}{\left(-\frac{x + 4}{y}\right) + \left(-\left(-\frac{x}{y} \cdot z\right)\right)}\right| \]
      sub-neg [<=] 91.2	\[ \left|\color{blue}{\left(-\frac{x + 4}{y}\right) - \left(-\frac{x}{y} \cdot z\right)}\right| \]
      distribute-neg-frac [=>] 91.2	\[ \left|\color{blue}{\frac{-\left(x + 4\right)}{y}} - \left(-\frac{x}{y} \cdot z\right)\right| \]
      associate-*l/ [=>] 99.9	\[ \left|\frac{-\left(x + 4\right)}{y} - \left(-\color{blue}{\frac{x \cdot z}{y}}\right)\right| \]
      distribute-neg-frac [=>] 99.9	\[ \left|\frac{-\left(x + 4\right)}{y} - \color{blue}{\frac{-x \cdot z}{y}}\right| \]
      neg-mul-1 [=>] 99.9	\[ \left|\frac{\color{blue}{-1 \cdot \left(x + 4\right)}}{y} - \frac{-x \cdot z}{y}\right| \]
      associate-*l/ [<=] 99.7	\[ \left|\color{blue}{\frac{-1}{y} \cdot \left(x + 4\right)} - \frac{-x \cdot z}{y}\right| \]
      neg-mul-1 [=>] 99.7	\[ \left|\frac{-1}{y} \cdot \left(x + 4\right) - \frac{\color{blue}{-1 \cdot \left(x \cdot z\right)}}{y}\right| \]
      associate-*l/ [<=] 99.7	\[ \left|\frac{-1}{y} \cdot \left(x + 4\right) - \color{blue}{\frac{-1}{y} \cdot \left(x \cdot z\right)}\right| \]
      distribute-lft-out-- [=>] 99.6	\[ \left|\color{blue}{\frac{-1}{y} \cdot \left(\left(x + 4\right) - x \cdot z\right)}\right| \]
      fabs-mul [=>] 99.6	\[ \color{blue}{\left|\frac{-1}{y}\right| \cdot \left|\left(x + 4\right) - x \cdot z\right|} \]
      fabs-sub [=>] 99.6	\[ \left|\frac{-1}{y}\right| \cdot \color{blue}{\left|x \cdot z - \left(x + 4\right)\right|} \]
      fabs-mul [<=] 99.6	\[ \color{blue}{\left|\frac{-1}{y} \cdot \left(x \cdot z - \left(x + 4\right)\right)\right|} \]
      associate-*l/ [=>] 99.9	\[ \left|\color{blue}{\frac{-1 \cdot \left(x \cdot z - \left(x + 4\right)\right)}{y}}\right| \]

   if 3.9999999999999999e-19 < (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z))) < 5.0000000000000001e291

   1. Initial program 99.8%

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

   if 5.0000000000000001e291 < (fabs.f64 (-.f64 (/.f64 (+.f64 x 4) y) (*.f64 (/.f64 x y) z)))

   1. Initial program 75.5%

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

   2. Simplified 100.0%

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

      [Start] 75.5	\[ \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \]
      fabs-neg [<=] 75.5	\[ \color{blue}{\left|-\left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)\right|} \]
      sub-neg [=>] 75.5	\[ \left|-\color{blue}{\left(\frac{x + 4}{y} + \left(-\frac{x}{y} \cdot z\right)\right)}\right| \]
      distribute-neg-in [=>] 75.5	\[ \left|\color{blue}{\left(-\frac{x + 4}{y}\right) + \left(-\left(-\frac{x}{y} \cdot z\right)\right)}\right| \]
      sub-neg [<=] 75.5	\[ \left|\color{blue}{\left(-\frac{x + 4}{y}\right) - \left(-\frac{x}{y} \cdot z\right)}\right| \]
      distribute-neg-frac [=>] 75.5	\[ \left|\color{blue}{\frac{-\left(x + 4\right)}{y}} - \left(-\frac{x}{y} \cdot z\right)\right| \]
      associate-*l/ [=>] 91.8	\[ \left|\frac{-\left(x + 4\right)}{y} - \left(-\color{blue}{\frac{x \cdot z}{y}}\right)\right| \]
      distribute-neg-frac [=>] 91.8	\[ \left|\frac{-\left(x + 4\right)}{y} - \color{blue}{\frac{-x \cdot z}{y}}\right| \]
      neg-mul-1 [=>] 91.8	\[ \left|\frac{\color{blue}{-1 \cdot \left(x + 4\right)}}{y} - \frac{-x \cdot z}{y}\right| \]
      associate-*l/ [<=] 91.8	\[ \left|\color{blue}{\frac{-1}{y} \cdot \left(x + 4\right)} - \frac{-x \cdot z}{y}\right| \]
      neg-mul-1 [=>] 91.8	\[ \left|\frac{-1}{y} \cdot \left(x + 4\right) - \frac{\color{blue}{-1 \cdot \left(x \cdot z\right)}}{y}\right| \]
      associate-*l/ [<=] 91.8	\[ \left|\frac{-1}{y} \cdot \left(x + 4\right) - \color{blue}{\frac{-1}{y} \cdot \left(x \cdot z\right)}\right| \]
      distribute-lft-out-- [=>] 100.0	\[ \left|\color{blue}{\frac{-1}{y} \cdot \left(\left(x + 4\right) - x \cdot z\right)}\right| \]
      fabs-mul [=>] 100.0	\[ \color{blue}{\left|\frac{-1}{y}\right| \cdot \left|\left(x + 4\right) - x \cdot z\right|} \]
      fabs-sub [=>] 100.0	\[ \left|\frac{-1}{y}\right| \cdot \color{blue}{\left|x \cdot z - \left(x + 4\right)\right|} \]
      fabs-mul [<=] 100.0	\[ \color{blue}{\left|\frac{-1}{y} \cdot \left(x \cdot z - \left(x + 4\right)\right)\right|} \]
      associate-*l/ [=>] 100.0	\[ \left|\color{blue}{\frac{-1 \cdot \left(x \cdot z - \left(x + 4\right)\right)}{y}}\right| \]

   3. Taylor expanded in x around 0 100.0%

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

   4. Simplified 100.0%

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

      [Start] 100.0	\[ \left|4 \cdot \frac{1}{y} + \left(\frac{1}{y} - \frac{z}{y}\right) \cdot x\right| \]
      associate-*r/ [=>] 100.0	\[ \left|\color{blue}{\frac{4 \cdot 1}{y}} + \left(\frac{1}{y} - \frac{z}{y}\right) \cdot x\right| \]
      metadata-eval [=>] 100.0	\[ \left|\frac{\color{blue}{4}}{y} + \left(\frac{1}{y} - \frac{z}{y}\right) \cdot x\right| \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \leq 4 \cdot 10^{-19}:\\ \;\;\;\;\left|\frac{x - \mathsf{fma}\left(x, z, -4\right)}{y}\right|\\ \mathbf{elif}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)\right|\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	21576

\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ t_1 := \left|t_0 - \frac{x}{y} \cdot z\right|\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{+82}:\\ \;\;\;\;\left|t_0 - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+291}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} + x \cdot \left(\frac{1}{y} - \frac{z}{y}\right)\right|\\ \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	21449

\[\begin{array}{l} t_0 := \left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-19} \lor \neg \left(t_0 \leq 10^{+269}\right):\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	99.2%
Cost	21448

\[\begin{array}{l} t_0 := \frac{x + 4}{y}\\ t_1 := \left|t_0 - \frac{x}{y} \cdot z\right|\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{+82}:\\ \;\;\;\;\left|t_0 - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;t_1 \leq 10^{+269}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array} \]

Alternative 4
Accuracy	68.2%
Cost	7380

\[\begin{array}{l} t_0 := \left|x \cdot \frac{z}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{+255}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-32}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	68.1%
Cost	7380

\[\begin{array}{l} t_0 := \left|x \cdot \frac{z}{y}\right|\\ t_1 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+254}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.12 \cdot 10^{+113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-31}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{+78}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	68.4%
Cost	7380

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -6 \cdot 10^{+233}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-68}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-32}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+92}:\\ \;\;\;\;\left|\frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	68.4%
Cost	7380

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -2.55 \cdot 10^{+241}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-68}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-32}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+75}:\\ \;\;\;\;\left|\frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	96.2%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;z \leq -3000000000:\\ \;\;\;\;\left|\frac{4 - x \cdot z}{y}\right|\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-7}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 9
Accuracy	95.3%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -3000000000 \lor \neg \left(z \leq 7.5 \cdot 10^{-7}\right):\\ \;\;\;\;\left|\frac{4 - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \end{array} \]

Alternative 10
Accuracy	96.0%
Cost	7104

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

Alternative 11
Accuracy	85.8%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+63}:\\ \;\;\;\;\left|x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+26}:\\ \;\;\;\;\left|\frac{-4 - x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{z}{\frac{y}{x}}\right|\\ \end{array} \]

Alternative 12
Accuracy	96.0%
Cost	6976

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

Alternative 13
Accuracy	69.7%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -1.52 \lor \neg \left(x \leq 4\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y}\right|\\ \end{array} \]

Alternative 14
Accuracy	40.7%
Cost	6592

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

Reproduce

herbie shell --seed 2023158 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))