Average Error: 5.9 → 0.1
Time: 10.9s
Precision: binary64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
↓
\[\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right) - 4 \cdot \left(z \cdot \left(y \cdot z\right)\right) \]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\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 x x (* 4.0 (* y t))) (* 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(x, x, (4.0 * (y * t))) - (4.0 * (z * (y * z)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original5.9
Target5.9
Herbie0.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \]

Derivation

  1. Initial program 5.9

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

  2. Simplified6.0

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

  3. Taylor expanded in y around 0 5.9

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

  4. Applied unpow2_binary645.9

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

  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 add-cube-cbrt_binary641.0

    \[\leadsto \color{blue}{\left(\sqrt[3]{4 \cdot \left(y \cdot t\right) + {x}^{2}} \cdot \sqrt[3]{4 \cdot \left(y \cdot t\right) + {x}^{2}}\right) \cdot \sqrt[3]{4 \cdot \left(y \cdot t\right) + {x}^{2}}} - 4 \cdot \left(\left(y \cdot z\right) \cdot z\right) \]

  7. Simplified1.0

    \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)}\right)} \cdot \sqrt[3]{4 \cdot \left(y \cdot t\right) + {x}^{2}} - 4 \cdot \left(\left(y \cdot z\right) \cdot z\right) \]

  8. Simplified1.0

    \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)}\right) \cdot \color{blue}{\sqrt[3]{\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot t\right)\right)}} - 4 \cdot \left(\left(y \cdot z\right) \cdot z\right) \]

  9. Applied rem-3cbrt-lft_binary640.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2021275 
(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))))