Average Error: 29.3 → 0.1
Time: 4.8s
Precision: binary64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
↓
\[-0.5 \cdot \mathsf{fma}\left(z, \frac{z + x}{y}, \left(-y\right) - \left(z + x\right) \cdot \frac{x}{y}\right) \]
(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))
↓
(FPCore (x y z)
 :precision binary64
 (* -0.5 (fma z (/ (+ z x) y) (- (- y) (* (+ z x) (/ x 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 * fma(z, ((z + x) / y), (-y - ((z + x) * (x / y))));
}

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end
↓
function code(x, y, z)
	return Float64(-0.5 * fma(z, Float64(Float64(z + x) / y), Float64(Float64(-y) - Float64(Float64(z + x) * Float64(x / y)))))
end

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := N[(-0.5 * N[(z * N[(N[(z + x), $MachinePrecision] / y), $MachinePrecision] + N[((-y) - N[(N[(z + x), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
-0.5 \cdot \mathsf{fma}\left(z, \frac{z + x}{y}, \left(-y\right) - \left(z + x\right) \cdot \frac{x}{y}\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original29.3
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 29.3

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

  2. Simplified0.1

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

  3. Taylor expanded in x around 0 13.2

    \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{{z}^{2}}{y} - \left(y + \frac{{x}^{2}}{y}\right)\right)} \]

  4. Simplified0.1

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

  5. Applied egg-rr0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2022150 
(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)))