Average Error: 2.8 → 0.1
Time: 7.1s
Precision: binary64
Cost: 13376
\[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
↓
\[x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)} \]
(FPCore (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))))
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));
}

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

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]

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)}

Error

Target

Original2.8
Target0.1
Herbie0.1
\[x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x} \]

Derivation

  1. Initial program 2.8

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

  2. Simplified0.1

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

    [Start]2.8

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

    --rgt-identity [<=]2.8

    \[ \color{blue}{\left(x - 0\right)} + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

    associate-+l- [=>]2.8

    \[ \color{blue}{x - \left(0 - \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)} \]

    sub-neg [=>]2.8

    \[ \color{blue}{x + \left(-\left(0 - \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)\right)} \]

    +-lft-identity [<=]2.8

    \[ x + \left(-\left(0 - \color{blue}{\left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)}\right)\right) \]

    sub0-neg [=>]2.8

    \[ x + \left(-\color{blue}{\left(-\left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)\right)}\right) \]

    neg-mul-1 [=>]2.8

    \[ x + \left(-\color{blue}{-1 \cdot \left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)}\right) \]

    distribute-lft-neg-in [=>]2.8

    \[ x + \color{blue}{\left(--1\right) \cdot \left(0 + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\right)} \]

    +-lft-identity [=>]2.8

    \[ x + \left(--1\right) \cdot \color{blue}{\frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

    associate-*r/ [=>]2.8

    \[ x + \color{blue}{\frac{\left(--1\right) \cdot y}{1.1283791670955126 \cdot e^{z} - x \cdot y}} \]

    sub-neg [=>]2.8

    \[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{1.1283791670955126 \cdot e^{z} + \left(-x \cdot y\right)}} \]

    +-commutative [=>]2.8

    \[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{\left(-x \cdot y\right) + 1.1283791670955126 \cdot e^{z}}} \]

    neg-sub0 [=>]2.8

    \[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{\left(0 - x \cdot y\right)} + 1.1283791670955126 \cdot e^{z}} \]

    associate-+l- [=>]2.8

    \[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{0 - \left(x \cdot y - 1.1283791670955126 \cdot e^{z}\right)}} \]

    sub0-neg [=>]2.8

    \[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{-\left(x \cdot y - 1.1283791670955126 \cdot e^{z}\right)}} \]

    neg-mul-1 [=>]2.8

    \[ x + \frac{\left(--1\right) \cdot y}{\color{blue}{-1 \cdot \left(x \cdot y - 1.1283791670955126 \cdot e^{z}\right)}} \]

    times-frac [=>]2.8

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

    metadata-eval [=>]2.8

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

    metadata-eval [=>]2.8

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

    associate-*r/ [=>]2.8

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

    associate-/l* [=>]2.8

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

    sub-neg [=>]2.8

    \[ x + \frac{-1}{\frac{\color{blue}{x \cdot y + \left(-1.1283791670955126 \cdot e^{z}\right)}}{y}} \]

    +-commutative [=>]2.8

    \[ x + \frac{-1}{\frac{\color{blue}{\left(-1.1283791670955126 \cdot e^{z}\right) + x \cdot y}}{y}} \]

    cancel-sign-sub [<=]2.8

    \[ x + \frac{-1}{\frac{\color{blue}{\left(-1.1283791670955126 \cdot e^{z}\right) - \left(-x\right) \cdot y}}{y}} \]

    distribute-lft-neg-in [<=]2.8

    \[ x + \frac{-1}{\frac{\left(-1.1283791670955126 \cdot e^{z}\right) - \color{blue}{\left(-x \cdot y\right)}}{y}} \]

    div-sub [=>]2.8

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

    *-lft-identity [<=]2.8

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

    metadata-eval [<=]2.8

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

    times-frac [<=]2.8

    \[ x + \frac{-1}{\frac{-1.1283791670955126 \cdot e^{z}}{y} - \color{blue}{\frac{-1 \cdot \left(-x \cdot y\right)}{-1 \cdot y}}} \]

    neg-mul-1 [<=]2.8

    \[ x + \frac{-1}{\frac{-1.1283791670955126 \cdot e^{z}}{y} - \frac{-1 \cdot \left(-x \cdot y\right)}{\color{blue}{-y}}} \]

    associate-*l/ [<=]2.8

    \[ x + \frac{-1}{\frac{-1.1283791670955126 \cdot e^{z}}{y} - \color{blue}{\frac{-1}{-y} \cdot \left(-x \cdot y\right)}} \]

    *-commutative [<=]2.8

    \[ x + \frac{-1}{\frac{-1.1283791670955126 \cdot e^{z}}{y} - \color{blue}{\left(-x \cdot y\right) \cdot \frac{-1}{-y}}} \]

    distribute-lft-neg-in [=>]2.8

    \[ x + \frac{-1}{\frac{-1.1283791670955126 \cdot e^{z}}{y} - \color{blue}{\left(\left(-x\right) \cdot y\right)} \cdot \frac{-1}{-y}} \]

    associate-*l* [=>]0.1

    \[ x + \frac{-1}{\frac{-1.1283791670955126 \cdot e^{z}}{y} - \color{blue}{\left(-x\right) \cdot \left(y \cdot \frac{-1}{-y}\right)}} \]

    associate-*r/ [=>]0.0

    \[ x + \frac{-1}{\frac{-1.1283791670955126 \cdot e^{z}}{y} - \left(-x\right) \cdot \color{blue}{\frac{y \cdot -1}{-y}}} \]

    *-commutative [<=]0.0

    \[ x + \frac{-1}{\frac{-1.1283791670955126 \cdot e^{z}}{y} - \left(-x\right) \cdot \frac{\color{blue}{-1 \cdot y}}{-y}} \]

    neg-mul-1 [<=]0.0

    \[ x + \frac{-1}{\frac{-1.1283791670955126 \cdot e^{z}}{y} - \left(-x\right) \cdot \frac{\color{blue}{-y}}{-y}} \]

    *-inverses [=>]0.0

    \[ x + \frac{-1}{\frac{-1.1283791670955126 \cdot e^{z}}{y} - \left(-x\right) \cdot \color{blue}{1}} \]

    *-rgt-identity [=>]0.0

    \[ x + \frac{-1}{\frac{-1.1283791670955126 \cdot e^{z}}{y} - \color{blue}{\left(-x\right)}} \]

    sub-neg [=>]0.0

    \[ x + \frac{-1}{\color{blue}{\frac{-1.1283791670955126 \cdot e^{z}}{y} + \left(-\left(-x\right)\right)}} \]

    remove-double-neg [=>]0.0

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

    distribute-lft-neg-in [=>]0.0

    \[ x + \frac{-1}{\frac{\color{blue}{\left(-1.1283791670955126\right) \cdot e^{z}}}{y} + x} \]

    associate-/l* [=>]0.0

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

    associate-/r/ [=>]0.1

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

    metadata-eval [=>]0.1

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

    metadata-eval [<=]0.1

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

    associate-/r* [<=]0.1

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

    neg-mul-1 [<=]0.1

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

    *-commutative [<=]0.1

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

    fma-def [=>]0.1

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

    neg-mul-1 [=>]0.1

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

    associate-/r* [=>]0.1

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

    metadata-eval [=>]0.1

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

  3. Final simplification0.1

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

Alternatives

Alternative 1
Error0.6
Cost13640
\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 2 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1:\\ \;\;\;\;x + \frac{-1}{x + \frac{-1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 2
Error10.1
Cost1112
\[\begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{1.1283791670955126}\\ \mathbf{if}\;z \leq -3.05 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-215}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-271}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 3
Error10.1
Cost1112
\[\begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{1.1283791670955126}\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-215}:\\ \;\;\;\;x + \frac{y}{x \cdot \left(-y\right)}\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-272}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 4
Error16.3
Cost848
\[\begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+184}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+118}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 5
Error0.4
Cost840
\[\begin{array}{l} \mathbf{if}\;z \leq -76:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{-7}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 6
Error19.3
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -7.9 \cdot 10^{-118}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-153}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 7
Error20.2
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-168}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 8
Error19.5
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022364 
(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)))))