Average Error: 13.0 → 3.0
Time: 2.7s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\frac{x}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y}{y - z}\right)\right)}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\frac{x}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y}{y - z}\right)\right)}
double code(double x, double y, double z) {
	return ((x * (y - z)) / y);
}
↓
double code(double x, double y, double z) {
	return (x / log1p(expm1((y / (y - z)))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.0
Target3.1
Herbie3.0
\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.69397660138285259 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

  1. Initial program 13.0

    \[\frac{x \cdot \left(y - z\right)}{y}\]
  2. Using strategy rm

  3. Applied associate-/l*3.0

    \[\leadsto \color{blue}{\frac{x}{\frac{y}{y - z}}}\]
  4. Using strategy rm

  5. Applied log1p-expm1-u3.0

    \[\leadsto \frac{x}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y}{y - z}\right)\right)}}\]

  6. Final simplification3.0

    \[\leadsto \frac{x}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{y}{y - z}\right)\right)}\]

Reproduce

herbie shell --seed 2020102 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< z -2.060202331921739e+104) (- x (/ (* z x) y)) (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

  (/ (* x (- y z)) y))