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

Average Error: 7.9 → 0.8

Time: 6.4s

Precision: binary64

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(x \cdot \frac{y}{a}, 0.5, \frac{-4.5}{\frac{\frac{a}{z}}{t}}\right)\\ t_2 := x \cdot y + t \cdot \left(z \cdot -9\right)\\ \mathbf{if}\;t_2 \leq -6.561417581202769 \cdot 10^{+257}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 7.645243099750725 \cdot 10^{+215}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4.5, z \cdot t, x \cdot \left(y \cdot 0.5\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (fma (* x (/ y a)) 0.5 (/ -4.5 (/ (/ a z) t))))
        (t_2 (+ (* x y) (* t (* z -9.0)))))
   (if (<= t_2 -6.561417581202769e+257)
     t_1
     (if (<= t_2 7.645243099750725e+215)
       (/ (fma -4.5 (* z t) (* x (* y 0.5))) a)
       t_1))))

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 = fma((x * (y / a)), 0.5, (-4.5 / ((a / z) / t)));
	double t_2 = (x * y) + (t * (z * -9.0));
	double tmp;
	if (t_2 <= -6.561417581202769e+257) {
		tmp = t_1;
	} else if (t_2 <= 7.645243099750725e+215) {
		tmp = fma(-4.5, (z * t), (x * (y * 0.5))) / a;
	} else {
		tmp = t_1;
	}
	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 = fma(Float64(x * Float64(y / a)), 0.5, Float64(-4.5 / Float64(Float64(a / z) / t)))
	t_2 = Float64(Float64(x * y) + Float64(t * Float64(z * -9.0)))
	tmp = 0.0
	if (t_2 <= -6.561417581202769e+257)
		tmp = t_1;
	elseif (t_2 <= 7.645243099750725e+215)
		tmp = Float64(fma(-4.5, Float64(z * t), Float64(x * Float64(y * 0.5))) / a);
	else
		tmp = t_1;
	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[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(-4.5 / N[(N[(a / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -6.561417581202769e+257], t$95$1, If[LessEqual[t$95$2, 7.645243099750725e+215], N[(N[(-4.5 * N[(z * t), $MachinePrecision] + N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.9
Target	5.8
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -6.5614175812027687e257 or 7.64524309975072541e215 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 35.8

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

   2. Simplified 35.7

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

   3. Taylor expanded in t around 0 35.5

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

   4. Applied egg-rr 0.8

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

   if -6.5614175812027687e257 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 7.64524309975072541e215

   1. Initial program 0.8

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

   2. Simplified 0.8

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

   3. Taylor expanded in t around 0 0.8

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

   4. Taylor expanded in a around 0 0.8

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

   5. Simplified 0.8

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4.5, t \cdot z, x \cdot \left(0.5 \cdot y\right)\right)}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + t \cdot \left(z \cdot -9\right) \leq -6.561417581202769 \cdot 10^{+257}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{y}{a}, 0.5, \frac{-4.5}{\frac{\frac{a}{z}}{t}}\right)\\ \mathbf{elif}\;x \cdot y + t \cdot \left(z \cdot -9\right) \leq 7.645243099750725 \cdot 10^{+215}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4.5, z \cdot t, x \cdot \left(y \cdot 0.5\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{y}{a}, 0.5, \frac{-4.5}{\frac{\frac{a}{z}}{t}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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)))