Average Error: 5.8 → 0.1
Time: 4.5s
Precision: binary64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
↓
\[\mathsf{fma}\left(4, y \cdot t, {x}^{2}\right) - 4 \cdot \left(z \cdot \left(y \cdot z\right)\right) \]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\mathsf{fma}\left(4, y \cdot t, {x}^{2}\right) - 4 \cdot \left(z \cdot \left(y \cdot z\right)\right)

(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))
↓
(FPCore (x y z t)
 :precision binary64
 (- (fma 4.0 (* y t) (pow x 2.0)) (* 4.0 (* z (* y z)))))
double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}
↓
double code(double x, double y, double z, double t) {
	return fma(4.0, (y * t), pow(x, 2.0)) - (4.0 * (z * (y * z)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original5.8
Target5.8
Herbie0.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \]

Derivation

  1. Initial program 5.8

    \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

  2. Simplified5.9

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, 4 \cdot \left(t - z \cdot z\right), x \cdot x\right)} \]

  3. Taylor expanded in y around 0 5.8

    \[\leadsto \color{blue}{\left(4 \cdot \left(y \cdot t\right) + {x}^{2}\right) - 4 \cdot \left(y \cdot {z}^{2}\right)} \]

  4. Applied unpow2_binary645.8

    \[\leadsto \left(4 \cdot \left(y \cdot t\right) + {x}^{2}\right) - 4 \cdot \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

  5. Applied associate-*r*_binary640.1

    \[\leadsto \left(4 \cdot \left(y \cdot t\right) + {x}^{2}\right) - 4 \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)} \]

  6. Applied fma-def_binary640.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, y \cdot t, {x}^{2}\right)} - 4 \cdot \left(\left(y \cdot z\right) \cdot z\right) \]

  7. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(4, y \cdot t, {x}^{2}\right) - 4 \cdot \left(z \cdot \left(y \cdot z\right)\right) \]

Reproduce

herbie shell --seed 2022093 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))