Average Error: 10.7 → 1.7
Time: 2.5s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\mathsf{fma}\left(y, \frac{x}{z}, \frac{x}{z}\right) - x \]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\mathsf{fma}\left(y, \frac{x}{z}, \frac{x}{z}\right) - x

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z) :precision binary64 (- (fma y (/ x z) (/ x z)) x))
double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}
↓
double code(double x, double y, double z) {
	return fma(y, (x / z), (x / z)) - x;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.7
Target0.5
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Derivation

  1. Initial program 10.7

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

  2. Applied associate-/l*_binary643.2

    \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}} \]

  3. Applied div-inv_binary643.3

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

  4. Taylor expanded in z around 0 3.6

    \[\leadsto \color{blue}{\left(\frac{y \cdot x}{z} + \frac{x}{z}\right) - x} \]

  5. Simplified1.7

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

  6. Final simplification1.7

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

Reproduce

herbie shell --seed 2022160 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))