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

Average Error: 7.5 → 0.7

Time: 6.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(0.5, x \cdot \frac{y}{a}, t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\right)\\ t_2 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ t_3 := \frac{\mathsf{fma}\left(t, z \cdot -4.5, x \cdot \frac{y}{2}\right)}{a}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-180}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 10^{-313}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{+210}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot -4.5}{\frac{a}{z}} + \frac{y \cdot 0.5}{\frac{a}{x}}\\ \end{array} \]

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 0.5 (* x (/ y a)) (* t (* -4.5 (/ z a)))))
        (t_2 (- (* x y) (* (* z 9.0) t)))
        (t_3 (/ (fma t (* z -4.5) (* x (/ y 2.0))) a)))
   (if (<= t_2 -1e+170)
     t_1
     (if (<= t_2 -1e-180)
       t_3
       (if (<= t_2 1e-313)
         t_1
         (if (<= t_2 1e+210)
           t_3
           (+ (/ (* t -4.5) (/ a z)) (/ (* y 0.5) (/ a x)))))))))

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(0.5, (x * (y / a)), (t * (-4.5 * (z / a))));
	double t_2 = (x * y) - ((z * 9.0) * t);
	double t_3 = fma(t, (z * -4.5), (x * (y / 2.0))) / a;
	double tmp;
	if (t_2 <= -1e+170) {
		tmp = t_1;
	} else if (t_2 <= -1e-180) {
		tmp = t_3;
	} else if (t_2 <= 1e-313) {
		tmp = t_1;
	} else if (t_2 <= 1e+210) {
		tmp = t_3;
	} else {
		tmp = ((t * -4.5) / (a / z)) + ((y * 0.5) / (a / x));
	}
	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(0.5, Float64(x * Float64(y / a)), Float64(t * Float64(-4.5 * Float64(z / a))))
	t_2 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	t_3 = Float64(fma(t, Float64(z * -4.5), Float64(x * Float64(y / 2.0))) / a)
	tmp = 0.0
	if (t_2 <= -1e+170)
		tmp = t_1;
	elseif (t_2 <= -1e-180)
		tmp = t_3;
	elseif (t_2 <= 1e-313)
		tmp = t_1;
	elseif (t_2 <= 1e+210)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(t * -4.5) / Float64(a / z)) + Float64(Float64(y * 0.5) / Float64(a / x)));
	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[(0.5 * N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] + N[(t * N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * N[(z * -4.5), $MachinePrecision] + N[(x * N[(y / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+170], t$95$1, If[LessEqual[t$95$2, -1e-180], t$95$3, If[LessEqual[t$95$2, 1e-313], t$95$1, If[LessEqual[t$95$2, 1e+210], t$95$3, N[(N[(N[(t * -4.5), $MachinePrecision] / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(N[(y * 0.5), $MachinePrecision] / N[(a / x), $MachinePrecision]), $MachinePrecision]), $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	5.8
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -1.00000000000000003e170 or -1e-180 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 1.00000000001e-313

   1. Initial program 18.9

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

   2. Simplified 18.9

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

   3. Taylor expanded in t around 0 18.8

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

   4. Applied egg-rr 11.0

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

   5. Applied egg-rr 1.7

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

   if -1.00000000000000003e170 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -1e-180 or 1.00000000001e-313 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 9.99999999999999927e209

   1. Initial program 0.3

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

   2. Simplified 0.3

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

   if 9.99999999999999927e209 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 29.7

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

   2. Simplified 29.5

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

   3. Taylor expanded in t around 0 29.4

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

   4. Applied egg-rr 16.2

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

   5. Applied egg-rr 1.0

      \[\leadsto \frac{t \cdot -4.5}{\frac{a}{z}} + \color{blue}{\frac{y \cdot 0.5}{\frac{a}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -1 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{y}{a}, t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -1 \cdot 10^{-180}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z \cdot -4.5, x \cdot \frac{y}{2}\right)}{a}\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 10^{-313}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \frac{y}{a}, t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 10^{+210}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z \cdot -4.5, x \cdot \frac{y}{2}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot -4.5}{\frac{a}{z}} + \frac{y \cdot 0.5}{\frac{a}{x}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022170 
(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)))