Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 13.1 → 0.3

Time: 6.6s

Precision: binary64

Cost: 2513

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{+56} \lor \neg \left(t_0 \leq 5 \cdot 10^{-49}\right) \land t_0 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (+ y z)) z)))
   (if (<= t_0 (- INFINITY))
     (/ x (/ z (+ y z)))
     (if (or (<= t_0 -1e+56) (and (not (<= t_0 5e-49)) (<= t_0 2e+299)))
       t_0
       (+ x (* x (/ y z)))))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = (x * (y + z)) / z;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x / (z / (y + z));
	} else if ((t_0 <= -1e+56) || (!(t_0 <= 5e-49) && (t_0 <= 2e+299))) {
		tmp = t_0;
	} else {
		tmp = x + (x * (y / z));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) / z;
}

↓

public static double code(double x, double y, double z) {
	double t_0 = (x * (y + z)) / z;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = x / (z / (y + z));
	} else if ((t_0 <= -1e+56) || (!(t_0 <= 5e-49) && (t_0 <= 2e+299))) {
		tmp = t_0;
	} else {
		tmp = x + (x * (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x * (y + z)) / z

↓

def code(x, y, z):
	t_0 = (x * (y + z)) / z
	tmp = 0
	if t_0 <= -math.inf:
		tmp = x / (z / (y + z))
	elif (t_0 <= -1e+56) or (not (t_0 <= 5e-49) and (t_0 <= 2e+299)):
		tmp = t_0
	else:
		tmp = x + (x * (y / z))
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(Float64(x * Float64(y + z)) / z)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x / Float64(z / Float64(y + z)));
	elseif ((t_0 <= -1e+56) || (!(t_0 <= 5e-49) && (t_0 <= 2e+299)))
		tmp = t_0;
	else
		tmp = Float64(x + Float64(x * Float64(y / z)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = (x * (y + z)) / z;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = x / (z / (y + z));
	elseif ((t_0 <= -1e+56) || (~((t_0 <= 5e-49)) && (t_0 <= 2e+299)))
		tmp = t_0;
	else
		tmp = x + (x * (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x / N[(z / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -1e+56], And[N[Not[LessEqual[t$95$0, 5e-49]], $MachinePrecision], LessEqual[t$95$0, 2e+299]]], t$95$0, N[(x + N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	13.1
Target	3.2
Herbie	0.3

\[\frac{x}{\frac{z}{y + z}} \]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y + z}}} \]
      Proof

      [Start] 64.0	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-/l* [=>] 0.1	\[ \color{blue}{\frac{x}{\frac{z}{y + z}}} \]

   if -inf.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < -1.00000000000000009e56 or 4.9999999999999999e-49 < (/.f64 (*.f64 x (+.f64 y z)) z) < 2.0000000000000001e299

   1. Initial program 0.2

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

   if -1.00000000000000009e56 < (/.f64 (*.f64 x (+.f64 y z)) z) < 4.9999999999999999e-49 or 2.0000000000000001e299 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 15.5

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{x \cdot \frac{y + z}{z}} \]
      Proof

      [Start] 15.5	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*r/ [<=] 0.3	\[ \color{blue}{x \cdot \frac{y + z}{z}} \]

   3. Taylor expanded in x around 0 15.5

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

   4. Simplified 0.3

      \[\leadsto \color{blue}{x + \frac{y}{z} \cdot x} \]
      Proof

      [Start] 15.5	\[ \frac{\left(y + z\right) \cdot x}{z} \]
      associate-*l/ [<=] 0.3	\[ \color{blue}{\frac{y + z}{z} \cdot x} \]
      *-lft-identity [<=] 0.3	\[ \color{blue}{\left(1 \cdot \frac{y + z}{z}\right)} \cdot x \]
      associate-*r/ [=>] 0.3	\[ \color{blue}{\frac{1 \cdot \left(y + z\right)}{z}} \cdot x \]
      associate-*l/ [<=] 0.4	\[ \color{blue}{\left(\frac{1}{z} \cdot \left(y + z\right)\right)} \cdot x \]
      distribute-lft-in [=>] 0.4	\[ \color{blue}{\left(\frac{1}{z} \cdot y + \frac{1}{z} \cdot z\right)} \cdot x \]
      lft-mult-inverse [=>] 0.3	\[ \left(\frac{1}{z} \cdot y + \color{blue}{1}\right) \cdot x \]
      distribute-rgt1-in [<=] 0.3	\[ \color{blue}{x + \left(\frac{1}{z} \cdot y\right) \cdot x} \]
      associate-*l/ [=>] 0.3	\[ x + \color{blue}{\frac{1 \cdot y}{z}} \cdot x \]
      *-lft-identity [=>] 0.3	\[ x + \frac{\color{blue}{y}}{z} \cdot x \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -1 \cdot 10^{+56} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \leq 5 \cdot 10^{-49}\right) \land \frac{x \cdot \left(y + z\right)}{z} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.3
Cost	1241

\[\begin{array}{l} t_0 := x \cdot \frac{y + z}{z}\\ t_1 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -2.25 \cdot 10^{-208}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-299}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-244}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-160} \lor \neg \left(z \leq 6.5 \cdot 10^{-111}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	3.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+14} \lor \neg \left(y \leq 6.1 \cdot 10^{+91}\right):\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array} \]

Alternative 3
Error	1.8
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-8} \lor \neg \left(y \leq 2 \cdot 10^{+89}\right):\\ \;\;\;\;x + y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array} \]

Alternative 4
Error	1.8
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-7} \lor \neg \left(y \leq 2 \cdot 10^{+89}\right):\\ \;\;\;\;x + y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 5
Error	18.4
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+33} \lor \neg \left(y \leq 8.8 \cdot 10^{+139}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	25.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023031 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))