Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.4 → 2.0

Time: 2.4s

Precision: 64

Math

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

↓

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

C

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

↓

double code(double x, double y, double z, double t) {
	return fma((z / t), (y - x), x);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.4
Target	1.9
Herbie	2.0

\[\begin{array}{l} \mathbf{if}\;x \lt -9.0255111955330046 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.2750321637007147 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

1. Initial program 6.4

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

2. Simplified 6.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)}\]
3. Using strategy rm

4. Applied clear-num 6.4

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{t}{y - x}}}, z, x\right)\]
5. Using strategy rm

6. Applied fma-udef 6.4

   \[\leadsto \color{blue}{\frac{1}{\frac{t}{y - x}} \cdot z + x}\]

7. Simplified 5.7

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

9. Applied associate-/r/ 2.0

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

10. Applied fma-def 2.0

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

11. Final simplification 2.0

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

Reproduce

herbie shell --seed 2020092 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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