Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.7 → 2.0

Time: 5.2s

Precision: binary64

Cost: 7752

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+308}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{+106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \end{array} \]

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 -1e+308)
     (/ x (/ z (+ y z)))
     (if (<= t_0 -5e+106) t_0 (fma x (/ y z) x)))))

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 <= -1e+308) {
		tmp = x / (z / (y + z));
	} else if (t_0 <= -5e+106) {
		tmp = t_0;
	} else {
		tmp = fma(x, (y / z), x);
	}
	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 <= -1e+308)
		tmp = Float64(x / Float64(z / Float64(y + z)));
	elseif (t_0 <= -5e+106)
		tmp = t_0;
	else
		tmp = fma(x, Float64(y / z), x);
	end
	return 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, -1e+308], N[(x / N[(z / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -5e+106], t$95$0, N[(x * N[(y / z), $MachinePrecision] + x), $MachinePrecision]]]]

Target

Original	12.7
Target	2.9
Herbie	2.0

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -1e308

   1. Initial program 63.9

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

   2. Simplified 0.1

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

      [Start] 63.9	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-/l* [=>] 0.1	\[ \color{blue}{\frac{x}{\frac{z}{y + z}}} \]

   if -1e308 < (/.f64 (*.f64 x (+.f64 y z)) z) < -4.9999999999999998e106

   1. Initial program 0.2

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

   if -4.9999999999999998e106 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 9.8

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

   2. Simplified 2.5

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

      [Start] 9.8	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*l/ [<=] 12.5	\[ \color{blue}{\frac{x}{z} \cdot \left(y + z\right)} \]
      distribute-rgt-in [=>] 12.5	\[ \color{blue}{y \cdot \frac{x}{z} + z \cdot \frac{x}{z}} \]
      associate-*r/ [=>] 14.7	\[ \color{blue}{\frac{y \cdot x}{z}} + z \cdot \frac{x}{z} \]
      associate-*l/ [<=] 13.3	\[ \color{blue}{\frac{y}{z} \cdot x} + z \cdot \frac{x}{z} \]
      *-commutative [=>] 13.3	\[ \color{blue}{x \cdot \frac{y}{z}} + z \cdot \frac{x}{z} \]
      *-commutative [=>] 13.3	\[ x \cdot \frac{y}{z} + \color{blue}{\frac{x}{z} \cdot z} \]
      associate-/r/ [<=] 2.5	\[ x \cdot \frac{y}{z} + \color{blue}{\frac{x}{\frac{z}{z}}} \]
      *-inverses [=>] 2.5	\[ x \cdot \frac{y}{z} + \frac{x}{\color{blue}{1}} \]
      /-rgt-identity [=>] 2.5	\[ x \cdot \frac{y}{z} + \color{blue}{x} \]
      fma-def [=>] 2.5	\[ \color{blue}{\mathsf{fma}\left(x, \frac{y}{z}, x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -1 \cdot 10^{+308}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -5 \cdot 10^{+106}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.0
Cost	1480

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+308}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{+106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2
Error	18.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+90} \lor \neg \left(y \leq 8.6 \cdot 10^{+122}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	19.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-106}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	19.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-111}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	3.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+99}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array} \]

Alternative 6
Error	3.2
Cost	448

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

Alternative 7
Error	3.2
Cost	448

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

Alternative 8
Error	2.9
Cost	448

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

Alternative 9
Error	25.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023060 
(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))