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

Average Accuracy: 88.2% → 99.5%

Time: 20.3s

Precision: binary64

Cost: 8392

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot \frac{0.5}{a}\right) - \frac{z}{\frac{a}{9}} \cdot \frac{t}{2}\\ t_2 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+269}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-320} \lor \neg \left(t_2 \leq 2 \cdot 10^{+303}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \end{array} \]

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 (- (* x (* y (/ 0.5 a))) (* (/ z (/ a 9.0)) (/ t 2.0))))
        (t_2 (- (* x y) (* (* z 9.0) t))))
   (if (<= t_2 -2e+269)
     t_1
     (if (<= t_2 -1e-285)
       (/ (fma z (* t -9.0) (* x y)) (* a 2.0))
       (if (or (<= t_2 5e-320) (not (<= t_2 2e+303)))
         t_1
         (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0)))))))

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 = (x * (y * (0.5 / a))) - ((z / (a / 9.0)) * (t / 2.0));
	double t_2 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_2 <= -2e+269) {
		tmp = t_1;
	} else if (t_2 <= -1e-285) {
		tmp = fma(z, (t * -9.0), (x * y)) / (a * 2.0);
	} else if ((t_2 <= 5e-320) || !(t_2 <= 2e+303)) {
		tmp = t_1;
	} else {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	}
	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(Float64(x * Float64(y * Float64(0.5 / a))) - Float64(Float64(z / Float64(a / 9.0)) * Float64(t / 2.0)))
	t_2 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_2 <= -2e+269)
		tmp = t_1;
	elseif (t_2 <= -1e-285)
		tmp = Float64(fma(z, Float64(t * -9.0), Float64(x * y)) / Float64(a * 2.0));
	elseif ((t_2 <= 5e-320) || !(t_2 <= 2e+303))
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(9.0 * t))) / Float64(a * 2.0));
	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 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(z / N[(a / 9.0), $MachinePrecision]), $MachinePrecision] * N[(t / 2.0), $MachinePrecision]), $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, -2e+269], t$95$1, If[LessEqual[t$95$2, -1e-285], N[(N[(z * N[(t * -9.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$2, 5e-320], N[Not[LessEqual[t$95$2, 2e+303]], $MachinePrecision]], t$95$1, N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	88.2%
Target	91.2%
Herbie	99.5%

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -2.0000000000000001e269 or -1.00000000000000007e-285 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 4.99994e-320 or 2e303 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 23.8%

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

   2. Applied egg-rr 61.1%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a} + \left(-\frac{z \cdot 9}{a} \cdot \frac{t}{2}\right)} \]
      Proof

      [Start] 23.8	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      div-sub [=>] 23.8	\[ \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]
      sub-neg [=>] 23.8	\[ \color{blue}{\frac{x \cdot y}{a \cdot 2} + \left(-\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right)} \]
      div-inv [=>] 23.8	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{a \cdot 2}} + \left(-\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right) \]
      *-commutative [=>] 23.8	\[ \left(x \cdot y\right) \cdot \frac{1}{\color{blue}{2 \cdot a}} + \left(-\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right) \]
      associate-/r* [=>] 23.8	\[ \left(x \cdot y\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} + \left(-\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right) \]
      metadata-eval [=>] 23.8	\[ \left(x \cdot y\right) \cdot \frac{\color{blue}{0.5}}{a} + \left(-\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right) \]
      times-frac [=>] 61.1	\[ \left(x \cdot y\right) \cdot \frac{0.5}{a} + \left(-\color{blue}{\frac{z \cdot 9}{a} \cdot \frac{t}{2}}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right) - \frac{z}{\frac{a}{9}} \cdot \frac{t}{2}} \]
      Proof

      [Start] 61.1	\[ \left(x \cdot y\right) \cdot \frac{0.5}{a} + \left(-\frac{z \cdot 9}{a} \cdot \frac{t}{2}\right) \]
      sub-neg [<=] 61.1	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a} - \frac{z \cdot 9}{a} \cdot \frac{t}{2}} \]
      associate-*l* [=>] 98.9	\[ \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right)} - \frac{z \cdot 9}{a} \cdot \frac{t}{2} \]
      associate-/l* [=>] 99.4	\[ x \cdot \left(y \cdot \frac{0.5}{a}\right) - \color{blue}{\frac{z}{\frac{a}{9}}} \cdot \frac{t}{2} \]

   if -2.0000000000000001e269 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -1.00000000000000007e-285

   1. Initial program 99.5%

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

   2. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}} \]
      Proof

      [Start] 99.5	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      sub-neg [=>] 99.5	\[ \frac{\color{blue}{x \cdot y + \left(-\left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2} \]
      +-commutative [=>] 99.5	\[ \frac{\color{blue}{\left(-\left(z \cdot 9\right) \cdot t\right) + x \cdot y}}{a \cdot 2} \]
      associate-*l* [=>] 99.5	\[ \frac{\left(-\color{blue}{z \cdot \left(9 \cdot t\right)}\right) + x \cdot y}{a \cdot 2} \]
      distribute-rgt-neg-in [=>] 99.5	\[ \frac{\color{blue}{z \cdot \left(-9 \cdot t\right)} + x \cdot y}{a \cdot 2} \]
      fma-def [=>] 99.5	\[ \frac{\color{blue}{\mathsf{fma}\left(z, -9 \cdot t, x \cdot y\right)}}{a \cdot 2} \]
      *-commutative [=>] 99.5	\[ \frac{\mathsf{fma}\left(z, -\color{blue}{t \cdot 9}, x \cdot y\right)}{a \cdot 2} \]
      distribute-rgt-neg-in [=>] 99.5	\[ \frac{\mathsf{fma}\left(z, \color{blue}{t \cdot \left(-9\right)}, x \cdot y\right)}{a \cdot 2} \]
      metadata-eval [=>] 99.5	\[ \frac{\mathsf{fma}\left(z, t \cdot \color{blue}{-9}, x \cdot y\right)}{a \cdot 2} \]

   if 4.99994e-320 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 2e303

   1. Initial program 99.5%

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

   2. Simplified 99.5%

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

      [Start] 99.5	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      associate-*l* [=>] 99.5	\[ \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -2 \cdot 10^{+269}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right) - \frac{z}{\frac{a}{9}} \cdot \frac{t}{2}\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right)}{a \cdot 2}\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{-320} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{+303}\right):\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right) - \frac{z}{\frac{a}{9}} \cdot \frac{t}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	3665

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot \frac{0.5}{a}\right) - \frac{z}{\frac{a}{9}} \cdot \frac{t}{2}\\ t_2 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+269}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-285}:\\ \;\;\;\;\frac{t_2}{a \cdot 2}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-320} \lor \neg \left(t_2 \leq 2 \cdot 10^{+303}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \end{array} \]

Alternative 2
Accuracy	90.0%
Cost	2384

\[\begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+266}:\\ \;\;\;\;z \cdot \frac{t}{\frac{a}{-4.5}}\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-143}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-180}:\\ \;\;\;\;\frac{y \cdot 0.5}{\frac{a}{x}}\\ \mathbf{elif}\;t_1 \leq 10^{+228}:\\ \;\;\;\;\frac{x \cdot y - t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \end{array} \]

Alternative 3
Accuracy	91.1%
Cost	2384

\[\begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-228}:\\ \;\;\;\;\frac{y}{a} \cdot \frac{x}{2} - 4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-180}:\\ \;\;\;\;\frac{y \cdot 0.5}{\frac{a}{x}}\\ \mathbf{elif}\;t_1 \leq 10^{+228}:\\ \;\;\;\;\frac{x \cdot y - t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \end{array} \]

Alternative 4
Accuracy	91.1%
Cost	2384

\[\begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-228}:\\ \;\;\;\;\frac{y}{a} \cdot \frac{x}{2} - \frac{\left(z \cdot t\right) \cdot 4.5}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-180}:\\ \;\;\;\;\frac{y \cdot 0.5}{\frac{a}{x}}\\ \mathbf{elif}\;t_1 \leq 10^{+228}:\\ \;\;\;\;\frac{x \cdot y - t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \end{array} \]

Alternative 5
Accuracy	91.1%
Cost	2384

\[\begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-228}:\\ \;\;\;\;\frac{\frac{y}{a}}{\frac{2}{x}} - \frac{\left(z \cdot t\right) \cdot 4.5}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-180}:\\ \;\;\;\;\frac{y \cdot 0.5}{\frac{a}{x}}\\ \mathbf{elif}\;t_1 \leq 10^{+228}:\\ \;\;\;\;\frac{x \cdot y - t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \end{array} \]

Alternative 6
Accuracy	87.3%
Cost	1493

\[\begin{array}{l} t_1 := \frac{0.5}{a} \cdot \left(x \cdot y + -9 \cdot \left(z \cdot t\right)\right)\\ \mathbf{if}\;t \leq -8.2 \cdot 10^{+14}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+15}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+224} \lor \neg \left(t \leq 1.9 \cdot 10^{+244}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 7
Accuracy	93.2%
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{y \cdot 0.5}{\frac{a}{x}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+248}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{a}{y} \cdot \frac{1}{x}}\\ \end{array} \]

Alternative 8
Accuracy	49.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{-239} \lor \neg \left(y \leq 1.25 \cdot 10^{-160}\right):\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 9
Accuracy	62.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-90} \lor \neg \left(t \leq 2.6 \cdot 10^{+52}\right):\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \end{array} \]

Alternative 10
Accuracy	62.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-91} \lor \neg \left(t \leq 1.15 \cdot 10^{+52}\right):\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \end{array} \]

Alternative 11
Accuracy	49.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-237}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-173}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \end{array} \]

Alternative 12
Accuracy	51.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+179}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-116}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]

Alternative 13
Accuracy	49.1%
Cost	448

\[-4.5 \cdot \left(t \cdot \frac{z}{a}\right) \]

Reproduce

herbie shell --seed 2023151 
(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)))