Average Error: 28.9 → 0.2
Time: 5.9s
Precision: binary64
Cost: 832
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[-0.5 \cdot \left(\left(z + x\right) \cdot \frac{z - x}{y} - y\right)\]
\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

Original28.9
Target0.2
Herbie0.2
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Alternatives

Alternative 1
Error0.2
Cost832
\[-0.5 \cdot \left(\frac{z + x}{\frac{y}{z - x}} - y\right)\]
Alternative 2
Error6.8
Cost904
\[\begin{array}{l} \mathbf{if}\;x \leq -1.4502890771219212 \cdot 10^{-15} \lor \neg \left(x \leq 724227.9020631777\right):\\ \;\;\;\;\left(y + x \cdot \frac{x}{y}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \end{array}\]
Alternative 3
Error9.6
Cost1169
\[\begin{array}{l} \mathbf{if}\;x \leq -1.3512482171332066 \cdot 10^{+154} \lor \neg \left(x \leq -7.292625313747265 \cdot 10^{-15} \lor \neg \left(x \leq 30.5176369781239\right) \land x \leq 1.0170312198005829 \cdot 10^{+144}\right):\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x \cdot x}{y}\right)\\ \end{array}\]
Alternative 4
Error14.5
Cost1169
\[\begin{array}{l} \mathbf{if}\;z \leq -3.7848748461932285 \cdot 10^{+86} \lor \neg \left(z \leq -2.458611196737199 \cdot 10^{-149} \lor \neg \left(z \leq 2.7611911349035228 \cdot 10^{-92}\right) \land z \leq 1.341976696412328 \cdot 10^{+154}\right):\\ \;\;\;\;0.5 \cdot \left(y + \frac{x \cdot x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z \cdot z}{y} - y\right)\\ \end{array}\]
Alternative 5
Error20.0
Cost576
\[-0.5 \cdot \left(\frac{z \cdot z}{y} - y\right)\]
Alternative 6
Error62.1
Cost64
\[1\]

Error

Derivation

  1. Initial program 28.9

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

  2. Simplified12.6

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

  4. Applied *-un-lft-identity_binary64_1337712.6

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

  5. Applied difference-of-squares_binary64_1334612.6

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

  6. Applied times-frac_binary64_133830.2

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

  7. Simplified0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2021044 
(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)))