Average Error: 0.5 → 0.4
Time: 7.0s
Precision: binary64
\[\log \left(1 + e^{x}\right) - x \cdot y \]
↓
\[\sqrt[3]{{\left(\mathsf{log1p}\left(e^{x}\right)\right)}^{3}} - x \cdot y \]
(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))
↓
(FPCore (x y) :precision binary64 (- (cbrt (pow (log1p (exp x)) 3.0)) (* x y)))
double code(double x, double y) {
	return log((1.0 + exp(x))) - (x * y);
}
↓
double code(double x, double y) {
	return cbrt(pow(log1p(exp(x)), 3.0)) - (x * y);
}

public static double code(double x, double y) {
	return Math.log((1.0 + Math.exp(x))) - (x * y);
}
↓
public static double code(double x, double y) {
	return Math.cbrt(Math.pow(Math.log1p(Math.exp(x)), 3.0)) - (x * y);
}

function code(x, y)
	return Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
end
↓
function code(x, y)
	return Float64(cbrt((log1p(exp(x)) ^ 3.0)) - Float64(x * y))
end

code[x_, y_] := N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[(N[Power[N[Power[N[Log[1 + N[Exp[x], $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]

\log \left(1 + e^{x}\right) - x \cdot y
↓
\sqrt[3]{{\left(\mathsf{log1p}\left(e^{x}\right)\right)}^{3}} - x \cdot y

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.5
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array} \]

Derivation

  1. Initial program 0.5

    \[\log \left(1 + e^{x}\right) - x \cdot y \]

  2. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]

  3. Applied egg-rr0.4

    \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(e^{x}\right)\right)}^{3}}} - x \cdot y \]

  4. Final simplification0.4

    \[\leadsto \sqrt[3]{{\left(\mathsf{log1p}\left(e^{x}\right)\right)}^{3}} - x \cdot y \]

Reproduce

herbie shell --seed 2022133 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1.0 (exp x))) (* x y)) (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y))))

  (- (log (+ 1.0 (exp x))) (* x y)))