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

Average Error: 3.7 → 0.9

Time: 5.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ t_2 := t_1 + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+303}:\\ \;\;\;\;\left(x + y \cdot \frac{-0.3333333333333333}{z}\right) + \frac{0.3333333333333333 \cdot \frac{t}{z}}{y}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+268}:\\ \;\;\;\;t_1 + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x + \frac{y}{z} \cdot -0.3333333333333333, \frac{\frac{t}{z}}{y \cdot 3}\right)\\ \end{array} \]

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 (- x (/ y (* z 3.0)))) (t_2 (+ t_1 (/ t (* y (* z 3.0))))))
   (if (<= t_2 -2e+303)
     (+
      (+ x (* y (/ -0.3333333333333333 z)))
      (/ (* 0.3333333333333333 (/ t z)) y))
     (if (<= t_2 2e+268)
       (+ t_1 (/ t (* z (* y 3.0))))
       (fma
        1.0
        (+ x (* (/ y z) -0.3333333333333333))
        (/ (/ t z) (* y 3.0)))))))

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 = x - (y / (z * 3.0));
	double t_2 = t_1 + (t / (y * (z * 3.0)));
	double tmp;
	if (t_2 <= -2e+303) {
		tmp = (x + (y * (-0.3333333333333333 / z))) + ((0.3333333333333333 * (t / z)) / y);
	} else if (t_2 <= 2e+268) {
		tmp = t_1 + (t / (z * (y * 3.0)));
	} else {
		tmp = fma(1.0, (x + ((y / z) * -0.3333333333333333)), ((t / z) / (y * 3.0)));
	}
	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(x - Float64(y / Float64(z * 3.0)))
	t_2 = Float64(t_1 + Float64(t / Float64(y * Float64(z * 3.0))))
	tmp = 0.0
	if (t_2 <= -2e+303)
		tmp = Float64(Float64(x + Float64(y * Float64(-0.3333333333333333 / z))) + Float64(Float64(0.3333333333333333 * Float64(t / z)) / y));
	elseif (t_2 <= 2e+268)
		tmp = Float64(t_1 + Float64(t / Float64(z * Float64(y * 3.0))));
	else
		tmp = fma(1.0, Float64(x + Float64(Float64(y / z) * -0.3333333333333333)), Float64(Float64(t / z) / Float64(y * 3.0)));
	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[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+303], N[(N[(x + N[(y * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(t / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+268], N[(t$95$1 + N[(t / N[(z * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(x + N[(N[(y / z), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(t / z), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	3.7
Target	1.9
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y))) < -2e303

   1. Initial program 49.8

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

   2. Applied egg-rr 1.5

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\left(t \cdot \frac{0.3333333333333333}{z}\right) \cdot \frac{1}{y}} \]

   3. Applied egg-rr 1.5

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t \cdot 0.3333333333333333}{z}}{y}} \]

   4. Applied egg-rr 1.5

      \[\leadsto \left(x - \color{blue}{\frac{0.3333333333333333}{z} \cdot y}\right) + \frac{\frac{t \cdot 0.3333333333333333}{z}}{y} \]

   5. Taylor expanded in t around 0 1.6

      \[\leadsto \left(x - \frac{0.3333333333333333}{z} \cdot y\right) + \frac{\color{blue}{0.3333333333333333 \cdot \frac{t}{z}}}{y} \]

   if -2e303 < (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y))) < 1.9999999999999999e268

   1. Initial program 0.5

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

   2. Applied egg-rr 0.5

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{{\left(z \cdot \left(3 \cdot y\right)\right)}^{1}}} \]

   if 1.9999999999999999e268 < (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y)))

   1. Initial program 17.3

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

   2. Applied egg-rr 3.9

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq -2 \cdot 10^{+303}:\\ \;\;\;\;\left(x + y \cdot \frac{-0.3333333333333333}{z}\right) + \frac{0.3333333333333333 \cdot \frac{t}{z}}{y}\\ \mathbf{elif}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 2 \cdot 10^{+268}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x + \frac{y}{z} \cdot -0.3333333333333333, \frac{\frac{t}{z}}{y \cdot 3}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))))