Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.2% → 96.3%

Time: 4.2s

Precision: binary64

Cost: 6852

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-161}:\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-161) (+ x (/ x (/ z y))) (fma (/ x z) y x)))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-161) {
		tmp = x + (x / (z / y));
	} else {
		tmp = fma((x / z), y, x);
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e-161)
		tmp = Float64(x + Float64(x / Float64(z / y)));
	else
		tmp = fma(Float64(x / z), y, x);
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_] := If[LessEqual[y, 5e-161], N[(x + N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y + x), $MachinePrecision]]

Target

Original	84.2%
Target	96.1%
Herbie	96.3%

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

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999999e-161

   1. Initial program 88.5%

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

   2. Simplified 96.3%

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

      [Start] 88.5%	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*l/ [<=] 79.8%	\[ \color{blue}{\frac{x}{z} \cdot \left(y + z\right)} \]
      distribute-rgt-in [=>] 75.6%	\[ \color{blue}{y \cdot \frac{x}{z} + z \cdot \frac{x}{z}} \]
      *-commutative [=>] 75.6%	\[ y \cdot \frac{x}{z} + \color{blue}{\frac{x}{z} \cdot z} \]
      associate-/r/ [<=] 89.7%	\[ y \cdot \frac{x}{z} + \color{blue}{\frac{x}{\frac{z}{z}}} \]
      *-inverses [=>] 89.7%	\[ y \cdot \frac{x}{z} + \frac{x}{\color{blue}{1}} \]
      /-rgt-identity [=>] 89.7%	\[ y \cdot \frac{x}{z} + \color{blue}{x} \]
      associate-*r/ [=>] 97.2%	\[ \color{blue}{\frac{y \cdot x}{z}} + x \]
      *-commutative [<=] 97.2%	\[ \frac{\color{blue}{x \cdot y}}{z} + x \]
      associate-*r/ [<=] 96.2%	\[ \color{blue}{x \cdot \frac{y}{z}} + x \]
      fma-def [=>] 96.3%	\[ \color{blue}{\mathsf{fma}\left(x, \frac{y}{z}, x\right)} \]

   3. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}} + x} \]
      Step-by-step derivation

      [Start] 96.3%	\[ \mathsf{fma}\left(x, \frac{y}{z}, x\right) \]
      fma-udef [=>] 96.2%	\[ \color{blue}{x \cdot \frac{y}{z} + x} \]
      clear-num [=>] 96.2%	\[ x \cdot \color{blue}{\frac{1}{\frac{z}{y}}} + x \]
      un-div-inv [=>] 96.4%	\[ \color{blue}{\frac{x}{\frac{z}{y}}} + x \]

   if 4.9999999999999999e-161 < y

   1. Initial program 79.2%

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

   2. Simplified 97.1%

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

      [Start] 79.2%	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*l/ [<=] 90.7%	\[ \color{blue}{\frac{x}{z} \cdot \left(y + z\right)} \]
      distribute-lft-in [=>] 83.8%	\[ \color{blue}{\frac{x}{z} \cdot y + \frac{x}{z} \cdot z} \]
      fma-def [=>] 83.8%	\[ \color{blue}{\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z} \cdot z\right)} \]
      associate-/r/ [<=] 97.1%	\[ \mathsf{fma}\left(\frac{x}{z}, y, \color{blue}{\frac{x}{\frac{z}{z}}}\right) \]
      *-inverses [=>] 97.1%	\[ \mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{\color{blue}{1}}\right) \]
      /-rgt-identity [=>] 97.1%	\[ \mathsf{fma}\left(\frac{x}{z}, y, \color{blue}{x}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-161}:\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.3%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-161}:\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array} \]

Alternative 2
Accuracy	72.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -21 \lor \neg \left(y \leq 1.05 \cdot 10^{-14}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	72.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -21.5:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4
Accuracy	94.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + z\right)\\ \end{array} \]

Alternative 5
Accuracy	95.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 50000000:\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + z\right)\\ \end{array} \]

Alternative 6
Accuracy	95.8%
Cost	448

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

Alternative 7
Accuracy	51.3%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023165 
(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))