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

Average Error: 7.6 → 0.7

Time: 7.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{-4.5}{\frac{\frac{a}{t}}{z}}\\ t_2 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_2 \leq -2.540492647148833 \cdot 10^{+263}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{x}{a}, 0.5, t_1\right)\\ \mathbf{elif}\;t_2 \leq -5.9208085336124974 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z \cdot -4.5, x \cdot \frac{y}{2}\right)}{a}\\ \mathbf{elif}\;t_2 \leq 4.265431290348826 \cdot 10^{-182}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{y}{a}, 0.5, \frac{-4.5}{\frac{\frac{a}{z}}{t}}\right)\\ \mathbf{elif}\;t_2 \leq 3.874140262510758 \cdot 10^{+182}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \frac{\left(z \cdot t\right) \cdot 4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\frac{a}{x}}, 0.5, t_1\right)\\ \end{array} \]

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 (/ -4.5 (/ (/ a t) z))) (t_2 (- (* x y) (* (* z 9.0) t))))
   (if (<= t_2 -2.540492647148833e+263)
     (fma (* y (/ x a)) 0.5 t_1)
     (if (<= t_2 -5.9208085336124974e-117)
       (/ (fma t (* z -4.5) (* x (/ y 2.0))) a)
       (if (<= t_2 4.265431290348826e-182)
         (fma (* x (/ y a)) 0.5 (/ -4.5 (/ (/ a z) t)))
         (if (<= t_2 3.874140262510758e+182)
           (- (* 0.5 (/ (* x y) a)) (/ (* (* z t) 4.5) a))
           (fma (/ y (/ a x)) 0.5 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 = -4.5 / ((a / t) / z);
	double t_2 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_2 <= -2.540492647148833e+263) {
		tmp = fma((y * (x / a)), 0.5, t_1);
	} else if (t_2 <= -5.9208085336124974e-117) {
		tmp = fma(t, (z * -4.5), (x * (y / 2.0))) / a;
	} else if (t_2 <= 4.265431290348826e-182) {
		tmp = fma((x * (y / a)), 0.5, (-4.5 / ((a / z) / t)));
	} else if (t_2 <= 3.874140262510758e+182) {
		tmp = (0.5 * ((x * y) / a)) - (((z * t) * 4.5) / a);
	} else {
		tmp = fma((y / (a / x)), 0.5, 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 = Float64(-4.5 / Float64(Float64(a / t) / z))
	t_2 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_2 <= -2.540492647148833e+263)
		tmp = fma(Float64(y * Float64(x / a)), 0.5, t_1);
	elseif (t_2 <= -5.9208085336124974e-117)
		tmp = Float64(fma(t, Float64(z * -4.5), Float64(x * Float64(y / 2.0))) / a);
	elseif (t_2 <= 4.265431290348826e-182)
		tmp = fma(Float64(x * Float64(y / a)), 0.5, Float64(-4.5 / Float64(Float64(a / z) / t)));
	elseif (t_2 <= 3.874140262510758e+182)
		tmp = Float64(Float64(0.5 * Float64(Float64(x * y) / a)) - Float64(Float64(Float64(z * t) * 4.5) / a));
	else
		tmp = fma(Float64(y / Float64(a / x)), 0.5, 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[(-4.5 / N[(N[(a / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2.540492647148833e+263], N[(N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, -5.9208085336124974e-117], N[(N[(t * N[(z * -4.5), $MachinePrecision] + N[(x * N[(y / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$2, 4.265431290348826e-182], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(-4.5 / N[(N[(a / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 3.874140262510758e+182], N[(N[(0.5 * N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(z * t), $MachinePrecision] * 4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.6
Target	5.6
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 5 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -2.5404926471488332e263

   1. Initial program 43.8

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

   2. Simplified 43.5

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

   3. Taylor expanded in t around 0 43.4

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

   4. Applied egg-rr 21.1

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{y}{1} \cdot \frac{x}{a}\right)} - 4.5 \cdot \frac{t \cdot z}{a} \]

   5. Applied egg-rr 0.3

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

   if -2.5404926471488332e263 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -5.9208085336124974e-117

   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 -5.9208085336124974e-117 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 4.2654312903488262e-182

   1. Initial program 5.1

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

   2. Simplified 5.1

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

   3. Taylor expanded in t around 0 5.1

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

   4. Applied egg-rr 2.2

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

   if 4.2654312903488262e-182 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 3.87414026251075791e182

   1. Initial program 0.3

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

   2. Simplified 0.3

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

   3. Taylor expanded in t around 0 0.3

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

   4. Applied egg-rr 0.3

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

   if 3.87414026251075791e182 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 26.3

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

   2. Simplified 26.2

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

   3. Taylor expanded in t around 0 26.1

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

   4. Applied egg-rr 14.9

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{y}{1} \cdot \frac{x}{a}\right)} - 4.5 \cdot \frac{t \cdot z}{a} \]

   5. Applied egg-rr 1.3

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

   6. Applied egg-rr 1.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{\frac{a}{x}}}, 0.5, \frac{-4.5}{\frac{\frac{a}{t}}{z}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -2.540492647148833 \cdot 10^{+263}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{x}{a}, 0.5, \frac{-4.5}{\frac{\frac{a}{t}}{z}}\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -5.9208085336124974 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z \cdot -4.5, x \cdot \frac{y}{2}\right)}{a}\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 4.265431290348826 \cdot 10^{-182}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{y}{a}, 0.5, \frac{-4.5}{\frac{\frac{a}{z}}{t}}\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 3.874140262510758 \cdot 10^{+182}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \frac{\left(z \cdot t\right) \cdot 4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\frac{a}{x}}, 0.5, \frac{-4.5}{\frac{\frac{a}{t}}{z}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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)))