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

Average Error: 3.6 → 1.5

Time: 4.6s

Precision: binary64

Math

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

↓

\[\begin{array}{l} t_1 := y - \frac{t}{y}\\ t_2 := x - \frac{y}{z \cdot 3}\\ t_3 := t_2 + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{+232}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, t_1, x\right)\\ \mathbf{elif}\;t_3 \leq 10^{+281}:\\ \;\;\;\;t_2 + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \frac{1}{z}, t_1, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- y (/ t y)))
        (t_2 (- x (/ y (* z 3.0))))
        (t_3 (+ t_2 (/ t (* y (* z 3.0))))))
   (if (<= t_3 -1e+232)
     (fma (/ -0.3333333333333333 z) t_1 x)
     (if (<= t_3 1e+281)
       (+ t_2 (/ t (* z (* y 3.0))))
       (fma (* -0.3333333333333333 (/ 1.0 z)) t_1 x)))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = y - (t / y);
	double t_2 = x - (y / (z * 3.0));
	double t_3 = t_2 + (t / (y * (z * 3.0)));
	double tmp;
	if (t_3 <= -1e+232) {
		tmp = fma((-0.3333333333333333 / z), t_1, x);
	} else if (t_3 <= 1e+281) {
		tmp = t_2 + (t / (z * (y * 3.0)));
	} else {
		tmp = fma((-0.3333333333333333 * (1.0 / z)), t_1, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

↓

function code(x, y, z, t)
	t_1 = Float64(y - Float64(t / y))
	t_2 = Float64(x - Float64(y / Float64(z * 3.0)))
	t_3 = Float64(t_2 + Float64(t / Float64(y * Float64(z * 3.0))))
	tmp = 0.0
	if (t_3 <= -1e+232)
		tmp = fma(Float64(-0.3333333333333333 / z), t_1, x);
	elseif (t_3 <= 1e+281)
		tmp = Float64(t_2 + Float64(t / Float64(z * Float64(y * 3.0))));
	else
		tmp = fma(Float64(-0.3333333333333333 * Float64(1.0 / z)), t_1, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+232], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * t$95$1 + x), $MachinePrecision], If[LessEqual[t$95$3, 1e+281], N[(t$95$2 + N[(t / N[(z * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * t$95$1 + x), $MachinePrecision]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	3.6
Target	1.9
Herbie	1.5

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y))) < -1.00000000000000006e232

   1. Initial program 11.2

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   2. Simplified 5.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y - \frac{t}{y}, x\right)} \]

   if -1.00000000000000006e232 < (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y))) < 1e281

   1. Initial program 0.5

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   2. Taylor expanded in z around 0 0.5

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{3 \cdot \left(y \cdot z\right)}} \]

   3. Simplified 0.5

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{z \cdot \left(y \cdot 3\right)}} \]

   if 1e281 < (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y)))

   1. Initial program 22.2

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   2. Simplified 4.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y - \frac{t}{y}, x\right)} \]

   3. Applied egg-rr 4.0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{z} \cdot -0.3333333333333333}, y - \frac{t}{y}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq -1 \cdot 10^{+232}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y - \frac{t}{y}, x\right)\\ \mathbf{elif}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 10^{+281}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \frac{1}{z}, y - \frac{t}{y}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))