Average Error: 28.8 → 0.1
Time: 6.7s
Precision: binary64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[-0.5 \cdot \left(\left(z + x\right) \cdot \frac{z - x}{y} - y\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
-0.5 \cdot \left(\left(z + x\right) \cdot \frac{z - x}{y} - y\right)
(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))
↓
(FPCore (x y z) :precision binary64 (* -0.5 (- (* (+ z x) (/ (- z x) y)) y)))
double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}
↓
double code(double x, double y, double z) {
	return -0.5 * (((z + x) * ((z - x) / y)) - y);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.8
Target0.2
Herbie0.1
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 28.8

    \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]

  2. Simplified12.7

    \[\leadsto \color{blue}{-0.5 \cdot \left(\frac{z \cdot z - x \cdot x}{y} - y\right)}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity_binary64_1133112.7

    \[\leadsto -0.5 \cdot \left(\frac{z \cdot z - x \cdot x}{\color{blue}{1 \cdot y}} - y\right)\]

  5. Applied difference-of-squares_binary64_1130012.7

    \[\leadsto -0.5 \cdot \left(\frac{\color{blue}{\left(z + x\right) \cdot \left(z - x\right)}}{1 \cdot y} - y\right)\]

  6. Applied times-frac_binary64_113370.1

    \[\leadsto -0.5 \cdot \left(\color{blue}{\frac{z + x}{1} \cdot \frac{z - x}{y}} - y\right)\]

  7. Simplified0.1

    \[\leadsto -0.5 \cdot \left(\color{blue}{\left(z + x\right)} \cdot \frac{z - x}{y} - y\right)\]

  8. Final simplification0.1

    \[\leadsto -0.5 \cdot \left(\left(z + x\right) \cdot \frac{z - x}{y} - y\right)\]

Reproduce

herbie shell --seed 2020354 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))