Average Error: 7.2 → 0.0
Time: 2.8s
Precision: binary64
\[\frac{x + y \cdot \left(z - x\right)}{z} \]
↓
\[\mathsf{fma}\left(\frac{x}{z}, 1 - y, y\right) \]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\mathsf{fma}\left(\frac{x}{z}, 1 - y, y\right)

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.2
Target3.8
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}} \]

Derivation

  1. Initial program 7.2

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

  2. Simplified7.2

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

  3. Applied *-un-lft-identity_binary647.2

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

  4. Applied *-un-lft-identity_binary647.2

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

  5. Applied times-frac_binary647.2

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

  6. Simplified7.2

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

  7. Simplified2.7

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

  8. Taylor expanded in y around 0 5.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(\frac{x}{z}, 1 - y, y\right) \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

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