Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Average Error: 0.0 → 0.0

Time: 17.1s

Precision: binary64

Math

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

↓

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

FPCore

(FPCore (x y z t) :precision binary64 (+ x (* (- y z) (- t x))))

↓

(FPCore (x y z t)
 :precision binary64
 (- (fma z x (fma t y x)) (fma z t (* x y))))

C

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

↓

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

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

3. Taylor expanded in y around 0 0.0

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

4. Applied add-cube-cbrt_binary64 0.5

   \[\leadsto \left(y \cdot t + \left(z \cdot x + x\right)\right) - \color{blue}{\left(\sqrt[3]{y \cdot x + t \cdot z} \cdot \sqrt[3]{y \cdot x + t \cdot z}\right) \cdot \sqrt[3]{y \cdot x + t \cdot z}} \]

5. Applied add-cube-cbrt_binary64 1.3

   \[\leadsto \color{blue}{\left(\sqrt[3]{y \cdot t + \left(z \cdot x + x\right)} \cdot \sqrt[3]{y \cdot t + \left(z \cdot x + x\right)}\right) \cdot \sqrt[3]{y \cdot t + \left(z \cdot x + x\right)}} - \left(\sqrt[3]{y \cdot x + t \cdot z} \cdot \sqrt[3]{y \cdot x + t \cdot z}\right) \cdot \sqrt[3]{y \cdot x + t \cdot z} \]

6. Applied prod-diff_binary64 1.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{y \cdot t + \left(z \cdot x + x\right)} \cdot \sqrt[3]{y \cdot t + \left(z \cdot x + x\right)}, \sqrt[3]{y \cdot t + \left(z \cdot x + x\right)}, -\sqrt[3]{y \cdot x + t \cdot z} \cdot \left(\sqrt[3]{y \cdot x + t \cdot z} \cdot \sqrt[3]{y \cdot x + t \cdot z}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{y \cdot x + t \cdot z}, \sqrt[3]{y \cdot x + t \cdot z} \cdot \sqrt[3]{y \cdot x + t \cdot z}, \sqrt[3]{y \cdot x + t \cdot z} \cdot \left(\sqrt[3]{y \cdot x + t \cdot z} \cdot \sqrt[3]{y \cdot x + t \cdot z}\right)\right)} \]

7. Simplified 0.0

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(z, x, \mathsf{fma}\left(t, y, x\right)\right) - \mathsf{fma}\left(z, t, x \cdot y\right)\right)} + \mathsf{fma}\left(-\sqrt[3]{y \cdot x + t \cdot z}, \sqrt[3]{y \cdot x + t \cdot z} \cdot \sqrt[3]{y \cdot x + t \cdot z}, \sqrt[3]{y \cdot x + t \cdot z} \cdot \left(\sqrt[3]{y \cdot x + t \cdot z} \cdot \sqrt[3]{y \cdot x + t \cdot z}\right)\right) \]

8. Simplified 0.0

   \[\leadsto \left(\mathsf{fma}\left(z, x, \mathsf{fma}\left(t, y, x\right)\right) - \mathsf{fma}\left(z, t, x \cdot y\right)\right) + \color{blue}{0} \]

9. Final simplification 0.0

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

Reproduce

herbie shell --seed 2022077 
(FPCore (x y z t)
  :name "Data.Metrics.Snapshot:quantile from metrics-0.3.0.2"
  :precision binary64

  :herbie-target
  (+ x (+ (* t (- y z)) (* (- x) (- y z))))

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