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

Average Error: 7.3 → 0.9

Time: 15.8s

Precision: binary64

Cost: 8520

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t}{\frac{a}{z}}\\ t_2 := x \cdot y + t \cdot \left(z \cdot -9\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-4.5, t_1, \frac{\frac{y \cdot 0.5}{a}}{\frac{1}{x}}\right)\\ \mathbf{elif}\;t_2 \leq 10^{+205}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4.5, t_1, 0.5 \cdot \frac{y}{\frac{a}{x}}\right)\\ \end{array} \]

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 (/ t (/ a z))) (t_2 (+ (* x y) (* t (* z -9.0)))))
   (if (<= t_2 (- INFINITY))
     (fma -4.5 t_1 (/ (/ (* y 0.5) a) (/ 1.0 x)))
     (if (<= t_2 1e+205)
       (/ (+ (* x y) (* z (* t -9.0))) (* a 2.0))
       (fma -4.5 t_1 (* 0.5 (/ y (/ 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 = t / (a / z);
	double t_2 = (x * y) + (t * (z * -9.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(-4.5, t_1, (((y * 0.5) / a) / (1.0 / x)));
	} else if (t_2 <= 1e+205) {
		tmp = ((x * y) + (z * (t * -9.0))) / (a * 2.0);
	} else {
		tmp = fma(-4.5, t_1, (0.5 * (y / (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 = Float64(t / Float64(a / z))
	t_2 = Float64(Float64(x * y) + Float64(t * Float64(z * -9.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(-4.5, t_1, Float64(Float64(Float64(y * 0.5) / a) / Float64(1.0 / x)));
	elseif (t_2 <= 1e+205)
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t * -9.0))) / Float64(a * 2.0));
	else
		tmp = fma(-4.5, t_1, Float64(0.5 * Float64(y / 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[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-4.5 * t$95$1 + N[(N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+205], N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-4.5 * t$95$1 + N[(0.5 * N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	7.3
Target	5.6
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 63.7

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

      [Start] 64.0	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      associate-*l* [=>] 63.7	\[ \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Taylor expanded in x around 0 63.5

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

   4. Simplified 0.3

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

      [Start] 63.5	\[ -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{y \cdot x}{a} \]
      fma-def [=>] 63.5	\[ \color{blue}{\mathsf{fma}\left(-4.5, \frac{t \cdot z}{a}, 0.5 \cdot \frac{y \cdot x}{a}\right)} \]
      associate-/l* [=>] 35.4	\[ \mathsf{fma}\left(-4.5, \color{blue}{\frac{t}{\frac{a}{z}}}, 0.5 \cdot \frac{y \cdot x}{a}\right) \]
      associate-/l* [=>] 0.3	\[ \mathsf{fma}\left(-4.5, \frac{t}{\frac{a}{z}}, 0.5 \cdot \color{blue}{\frac{y}{\frac{a}{x}}}\right) \]

   5. Applied egg-rr 0.4

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 1.00000000000000002e205

   1. Initial program 0.8

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

   2. Simplified 0.9

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

      [Start] 0.8	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      associate-*l* [=>] 0.9	\[ \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   if 1.00000000000000002e205 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 28.8

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

   2. Simplified 28.6

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

      [Start] 28.8	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      associate-*l* [=>] 28.6	\[ \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Taylor expanded in x around 0 28.4

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

   4. Simplified 1.2

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

      [Start] 28.4	\[ -4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{y \cdot x}{a} \]
      fma-def [=>] 28.4	\[ \color{blue}{\mathsf{fma}\left(-4.5, \frac{t \cdot z}{a}, 0.5 \cdot \frac{y \cdot x}{a}\right)} \]
      associate-/l* [=>] 16.2	\[ \mathsf{fma}\left(-4.5, \color{blue}{\frac{t}{\frac{a}{z}}}, 0.5 \cdot \frac{y \cdot x}{a}\right) \]
      associate-/l* [=>] 1.2	\[ \mathsf{fma}\left(-4.5, \frac{t}{\frac{a}{z}}, 0.5 \cdot \color{blue}{\frac{y}{\frac{a}{x}}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + t \cdot \left(z \cdot -9\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-4.5, \frac{t}{\frac{a}{z}}, \frac{\frac{y \cdot 0.5}{a}}{\frac{1}{x}}\right)\\ \mathbf{elif}\;x \cdot y + t \cdot \left(z \cdot -9\right) \leq 10^{+205}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4.5, \frac{t}{\frac{a}{z}}, 0.5 \cdot \frac{y}{\frac{a}{x}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	8520

\[\begin{array}{l} t_1 := x \cdot y + t \cdot \left(z \cdot -9\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+243}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} + -4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;t_1 \leq 10^{+205}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4.5, \frac{t}{\frac{a}{z}}, 0.5 \cdot \frac{y}{\frac{a}{x}}\right)\\ \end{array} \]

Alternative 2
Error	0.9
Cost	2249

\[\begin{array}{l} t_1 := x \cdot y + t \cdot \left(z \cdot -9\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+243} \lor \neg \left(t_1 \leq 10^{+238}\right):\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{y}{a} + -4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}\\ \end{array} \]

Alternative 3
Error	3.9
Cost	2120

\[\begin{array}{l} t_1 := x \cdot y + t \cdot \left(z \cdot -9\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{\frac{a}{0.5}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \]

Alternative 4
Error	25.3
Cost	1504

\[\begin{array}{l} t_1 := -4.5 \cdot \frac{z \cdot t}{a}\\ t_2 := 0.5 \cdot \frac{y}{\frac{a}{x}}\\ t_3 := 0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+103}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-19}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-87}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-119}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-221}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-187}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	24.5
Cost	1372

\[\begin{array}{l} t_1 := -4.5 \cdot \frac{z \cdot t}{a}\\ t_2 := 0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-87}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;x \leq -3.75 \cdot 10^{-119}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-256}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	25.7
Cost	1240

\[\begin{array}{l} t_1 := -4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-181}:\\ \;\;\;\;x \cdot \frac{y}{\frac{a}{0.5}}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-159}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \]

Alternative 7
Error	25.7
Cost	1240

\[\begin{array}{l} t_1 := z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{if}\;t \leq -2.75 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-180}:\\ \;\;\;\;x \cdot \frac{y}{\frac{a}{0.5}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-159}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \]

Alternative 8
Error	25.3
Cost	1240

\[\begin{array}{l} t_1 := z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{if}\;t \leq -2.75 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-181}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \]

Alternative 9
Error	25.3
Cost	1240

\[\begin{array}{l} t_1 := \frac{z}{\frac{\frac{a}{-4.5}}{t}}\\ \mathbf{if}\;t \leq -2.75 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-180}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-159}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+20}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \]

Alternative 10
Error	25.3
Cost	1240

\[\begin{array}{l} t_1 := \frac{z}{\frac{\frac{a}{-4.5}}{t}}\\ \mathbf{if}\;t \leq -2.75 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-180}:\\ \;\;\;\;\frac{x \cdot y}{\frac{a}{0.5}}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-159}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+22}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+85}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \end{array} \]

Alternative 11
Error	32.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-262}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-107}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \end{array} \]

Alternative 12
Error	32.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-263}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 13
Error	32.7
Cost	448

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

Reproduce

herbie shell --seed 2023046 
(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)))