Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, I

Average Error: 7.5 → 7.4

Time: 8.2s

Precision: binary64

Math

\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

↓

\[\frac{\mathsf{fma}\left(-4.5, t \cdot z, 0.5 \cdot \left(y \cdot x\right)\right)}{a} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

↓

(FPCore (x y z t a)
 :precision binary64
 (/ (fma -4.5 (* t z) (* 0.5 (* y x))) a))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

↓

double code(double x, double y, double z, double t, double a) {
	return fma(-4.5, (t * z), (0.5 * (y * x))) / a;
}

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

↓

function code(x, y, z, t, a)
	return Float64(fma(-4.5, Float64(t * z), Float64(0.5 * Float64(y * x))) / a)
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := N[(N[(-4.5 * N[(t * z), $MachinePrecision] + N[(0.5 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.5
Target	5.2
Herbie	7.4

\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \]

Derivation

1. Initial program 7.5

   \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

2. Taylor expanded in x around 0 7.4

   \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t \cdot z}{a}} \]

3. Simplified 7.4

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4.5, t \cdot z, 0.5 \cdot \left(y \cdot x\right)\right)}{a}} \]

4. Final simplification 7.4

   \[\leadsto \frac{\mathsf{fma}\left(-4.5, t \cdot z, 0.5 \cdot \left(y \cdot x\right)\right)}{a} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))