Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Accuracy: 79.8% → 99.4%

Time: 7.2s

Precision: binary64

Cost: 2513

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{+50} \lor \neg \left(t_0 \leq 2 \cdot 10^{-77}\right) \land t_0 \leq 5 \cdot 10^{+287}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \end{array} \]

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))
     (* (+ y z) (/ x z))
     (if (or (<= t_0 -5e+50) (and (not (<= t_0 2e-77)) (<= t_0 5e+287)))
       t_0
       (+ x (/ x (/ z y)))))))

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 = (y + z) * (x / z);
	} else if ((t_0 <= -5e+50) || (!(t_0 <= 2e-77) && (t_0 <= 5e+287))) {
		tmp = t_0;
	} else {
		tmp = x + (x / (z / y));
	}
	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 = (y + z) * (x / z);
	} else if ((t_0 <= -5e+50) || (!(t_0 <= 2e-77) && (t_0 <= 5e+287))) {
		tmp = t_0;
	} else {
		tmp = x + (x / (z / y));
	}
	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 = (y + z) * (x / z)
	elif (t_0 <= -5e+50) or (not (t_0 <= 2e-77) and (t_0 <= 5e+287)):
		tmp = t_0
	else:
		tmp = x + (x / (z / y))
	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(Float64(y + z) * Float64(x / z));
	elseif ((t_0 <= -5e+50) || (!(t_0 <= 2e-77) && (t_0 <= 5e+287)))
		tmp = t_0;
	else
		tmp = Float64(x + Float64(x / Float64(z / y)));
	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 = (y + z) * (x / z);
	elseif ((t_0 <= -5e+50) || (~((t_0 <= 2e-77)) && (t_0 <= 5e+287)))
		tmp = t_0;
	else
		tmp = x + (x / (z / y));
	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[(N[(y + z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, -5e+50], And[N[Not[LessEqual[t$95$0, 2e-77]], $MachinePrecision], LessEqual[t$95$0, 5e+287]]], t$95$0, N[(x + N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	79.8%
Target	95.0%
Herbie	99.4%

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 0.0%

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

   2. Simplified 99.8%

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

      [Start] 0.0	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*l/ [<=] 99.8	\[ \color{blue}{\frac{x}{z} \cdot \left(y + z\right)} \]

   if -inf.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < -5e50 or 1.9999999999999999e-77 < (/.f64 (*.f64 x (+.f64 y z)) z) < 5e287

   1. Initial program 99.6%

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

   if -5e50 < (/.f64 (*.f64 x (+.f64 y z)) z) < 1.9999999999999999e-77 or 5e287 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 75.7%

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

   2. Simplified 99.2%

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

      [Start] 75.7	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*l/ [<=] 73.6	\[ \color{blue}{\frac{x}{z} \cdot \left(y + z\right)} \]
      distribute-rgt-in [=>] 73.6	\[ \color{blue}{y \cdot \frac{x}{z} + z \cdot \frac{x}{z}} \]
      *-commutative [=>] 73.6	\[ y \cdot \frac{x}{z} + \color{blue}{\frac{x}{z} \cdot z} \]
      associate-/r/ [<=] 94.5	\[ y \cdot \frac{x}{z} + \color{blue}{\frac{x}{\frac{z}{z}}} \]
      *-inverses [=>] 94.5	\[ y \cdot \frac{x}{z} + \frac{x}{\color{blue}{1}} \]
      /-rgt-identity [=>] 94.5	\[ y \cdot \frac{x}{z} + \color{blue}{x} \]
      associate-*r/ [=>] 90.3	\[ \color{blue}{\frac{y \cdot x}{z}} + x \]
      *-commutative [<=] 90.3	\[ \frac{\color{blue}{x \cdot y}}{z} + x \]
      associate-*r/ [<=] 99.1	\[ \color{blue}{x \cdot \frac{y}{z}} + x \]
      fma-def [=>] 99.2	\[ \color{blue}{\mathsf{fma}\left(x, \frac{y}{z}, x\right)} \]

   3. Applied egg-rr 99.2%

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

      [Start] 99.2	\[ \mathsf{fma}\left(x, \frac{y}{z}, x\right) \]
      fma-udef [=>] 99.1	\[ \color{blue}{x \cdot \frac{y}{z} + x} \]
      clear-num [=>] 99.1	\[ x \cdot \color{blue}{\frac{1}{\frac{z}{y}}} + x \]
      un-div-inv [=>] 99.2	\[ \color{blue}{\frac{x}{\frac{z}{y}}} + x \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -\infty:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -5 \cdot 10^{+50} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \leq 2 \cdot 10^{-77}\right) \land \frac{x \cdot \left(y + z\right)}{z} \leq 5 \cdot 10^{+287}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	69.2%
Cost	849

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+102}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -5 \cdot 10^{+56} \lor \neg \left(z \leq -5.7 \cdot 10^{+19}\right) \land z \leq 5.6 \cdot 10^{-121}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	69.4%
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+102}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -5 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-121}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	69.4%
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+103}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.32 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq -6 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-122}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	68.2%
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+102}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -5 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-200}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	94.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-152} \lor \neg \left(z \leq -4.2 \cdot 10^{-217}\right):\\ \;\;\;\;x \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \end{array} \]

Alternative 6
Accuracy	95.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-152} \lor \neg \left(z \leq -4.8 \cdot 10^{-220}\right):\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \end{array} \]

Alternative 7
Accuracy	94.5%
Cost	448

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

Alternative 8
Accuracy	61.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023151 
(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))