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

Average Error: 7.7 → 1.3

Time: 11.4s

Precision: binary64

\[ \begin{array}{c}[z, t] = \mathsf{sort}([z, t])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, z \cdot \frac{-4.5}{\frac{a}{t}}\right)\\ \mathbf{elif}\;t_1 \leq 10^{+296}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 0.5, \frac{x}{a}, \frac{t}{a} \cdot \left(z \cdot -4.5\right)\right) + \mathsf{fma}\left(\frac{-t}{a}, z \cdot 4.5, \frac{t}{a} \cdot \left(z \cdot 4.5\right)\right)\\ \end{array} \]

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
 (let* ((t_1 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0))))
   (if (<= t_1 -1e+297)
     (fma (/ x a) (/ y 2.0) (* z (/ -4.5 (/ a t))))
     (if (<= t_1 1e+296)
       (/ (fma z (* t -9.0) (* x y)) (* a 2.0))
       (+
        (fma (* y 0.5) (/ x a) (* (/ t a) (* z -4.5)))
        (fma (/ (- t) a) (* z 4.5) (* (/ t a) (* z 4.5))))))))

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) {
	double t_1 = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
	double tmp;
	if (t_1 <= -1e+297) {
		tmp = fma((x / a), (y / 2.0), (z * (-4.5 / (a / t))));
	} else if (t_1 <= 1e+296) {
		tmp = fma(z, (t * -9.0), (x * y)) / (a * 2.0);
	} else {
		tmp = fma((y * 0.5), (x / a), ((t / a) * (z * -4.5))) + fma((-t / a), (z * 4.5), ((t / a) * (z * 4.5)));
	}
	return tmp;
}

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)
	t_1 = Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
	tmp = 0.0
	if (t_1 <= -1e+297)
		tmp = fma(Float64(x / a), Float64(y / 2.0), Float64(z * Float64(-4.5 / Float64(a / t))));
	elseif (t_1 <= 1e+296)
		tmp = Float64(fma(z, Float64(t * -9.0), Float64(x * y)) / Float64(a * 2.0));
	else
		tmp = Float64(fma(Float64(y * 0.5), Float64(x / a), Float64(Float64(t / a) * Float64(z * -4.5))) + fma(Float64(Float64(-t) / a), Float64(z * 4.5), Float64(Float64(t / a) * Float64(z * 4.5))));
	end
	return tmp
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_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+297], N[(N[(x / a), $MachinePrecision] * N[(y / 2.0), $MachinePrecision] + N[(z * N[(-4.5 / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+296], N[(N[(z * N[(t * -9.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * 0.5), $MachinePrecision] * N[(x / a), $MachinePrecision] + N[(N[(t / a), $MachinePrecision] * N[(z * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[((-t) / a), $MachinePrecision] * N[(z * 4.5), $MachinePrecision] + N[(N[(t / a), $MachinePrecision] * N[(z * 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.7
Target	5.6
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) (*.f64 a 2)) < -1e297

   1. Initial program 55.5

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

   2. Applied egg-rr 31.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, -\left(9 \cdot \left(z \cdot t\right)\right) \cdot \frac{0.5}{a}\right)} \]

   3. Taylor expanded in z around 0 31.5

      \[\leadsto \mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, -\color{blue}{4.5 \cdot \frac{t \cdot z}{a}}\right) \]

   4. Simplified 3.6

      \[\leadsto \mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, -\color{blue}{z \cdot \frac{4.5}{\frac{a}{t}}}\right) \]

   if -1e297 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) (*.f64 a 2)) < 9.99999999999999981e295

   1. Initial program 0.9

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

   2. Applied egg-rr 0.9

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, -9 \cdot t, x \cdot y\right)}}{a \cdot 2} \]

   if 9.99999999999999981e295 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) (*.f64 a 2))

   1. Initial program 56.8

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

   2. Applied egg-rr 30.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, -\left(9 \cdot \left(z \cdot t\right)\right) \cdot \frac{0.5}{a}\right)} \]

   3. Taylor expanded in z around 0 29.9

      \[\leadsto \mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, -\color{blue}{4.5 \cdot \frac{t \cdot z}{a}}\right) \]

   4. Simplified 3.3

      \[\leadsto \mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, -\color{blue}{z \cdot \frac{4.5}{\frac{a}{t}}}\right) \]

   5. Applied egg-rr 3.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot 0.5, \frac{x}{a}, -\frac{t}{a} \cdot \left(z \cdot 4.5\right)\right) + \mathsf{fma}\left(-\frac{t}{a}, z \cdot 4.5, \frac{t}{a} \cdot \left(z \cdot 4.5\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \leq -1 \cdot 10^{+297}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, z \cdot \frac{-4.5}{\frac{a}{t}}\right)\\ \mathbf{elif}\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \leq 10^{+296}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 0.5, \frac{x}{a}, \frac{t}{a} \cdot \left(z \cdot -4.5\right)\right) + \mathsf{fma}\left(\frac{-t}{a}, z \cdot 4.5, \frac{t}{a} \cdot \left(z \cdot 4.5\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022209 
(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)))