Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A

Average Error: 4.53% → 0.07%

Time: 12.1s

Precision: binary64

Cost: 13376

Math

\[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

↓

\[x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))

↓

(FPCore (x y z)
 :precision binary64
 (+ x (/ -1.0 (fma (exp z) (/ -1.1283791670955126 y) x))))

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}

↓

double code(double x, double y, double z) {
	return x + (-1.0 / fma(exp(z), (-1.1283791670955126 / y), x));
}

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

↓

function code(x, y, z)
	return Float64(x + Float64(-1.0 / fma(exp(z), Float64(-1.1283791670955126 / y), x)))
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(x + N[(-1.0 / N[(N[Exp[z], $MachinePrecision] * N[(-1.1283791670955126 / y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	4.53%
Target	0.07%
Herbie	0.07%

\[x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x} \]

Derivation

1. Initial program 4.53

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

2. Simplified 0.07

   \[\leadsto \color{blue}{x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)}} \]
   Proof

   [Start] 4.53	\[ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   --rgt-identity [<=] 4.53	\[ \color{blue}{\left(x - 0\right)} + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   associate-+l- [=>] 4.53	\[ \color{blue}{x - \left(0 - \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)} \]
   sub-neg [=>] 4.53	\[ \color{blue}{x + \left(-\left(0 - \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)\right)} \]
   +-lft-identity [<=] 4.53	\[ x + \left(-\left(0 - \color{blue}{\left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)}\right)\right) \]
   sub0-neg [=>] 4.53	\[ x + \left(-\color{blue}{\left(-\left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)\right)}\right) \]
   neg-mul-1 [=>] 4.53	\[ x + \left(-\color{blue}{-1 \cdot \left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)}\right) \]
   distribute-lft-neg-in [=>] 4.53	\[ x + \color{blue}{\left(--1\right) \cdot \left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)} \]
   +-lft-identity [=>] 4.53	\[ x + \left(--1\right) \cdot \color{blue}{\frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]
   associate-*r/ [=>] 4.53	\[ x + \color{blue}{\frac{\left(--1\right) \cdot y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]
   sub-neg [=>] 4.53	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{1.1283791670955126 \cdot e^{z} + \left(-x \cdot y\right)}} \]
   +-commutative [=>] 4.53	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{\left(-x \cdot y\right) + 1.1283791670955126 \cdot e^{z}}} \]
   neg-sub0 [=>] 4.53	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{\left(0 - x \cdot y\right)} + 1.1283791670955126 \cdot e^{z}} \]
   associate-+l- [=>] 4.53	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{0 - \left(x \cdot y - 1.1283791670955126 \cdot e^{z}\right)}} \]
   sub0-neg [=>] 4.53	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{-\left(x \cdot y - 1.1283791670955126 \cdot e^{z}\right)}} \]
   neg-mul-1 [=>] 4.53	\[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{-1 \cdot \left(x \cdot y - 1.1283791670955126 \cdot e^{z}\right)}} \]
   times-frac [=>] 4.53	\[ x + \color{blue}{\frac{--1}{-1} \cdot \frac{y}{x \cdot y - 1.1283791670955126 \cdot e^{z}}} \]

3. Final simplification 0.07

   \[\leadsto x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)} \]

Alternatives

Alternative 1
Error	0.09%
Cost	20168

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.29%
Cost	13896

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot \left(z + 1\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	13.41%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-109}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-25}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 + z \cdot 1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	0.54%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -7000000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 3.5:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	0.52%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -7000000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 3.5:\\ \;\;\;\;x + \frac{-1}{x + \frac{-1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	27.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.05 \cdot 10^{-56}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-173}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	13.42%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-109}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 3.65 \cdot 10^{-26}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	29.77%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 10^{-187}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	30.03%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-131}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-188}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	31.18%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))