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

Average Error: 7.5 → 4.2

Time: 11.3s

Precision: binary64

Cost: 8392

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{x}}{y \cdot 0.5}}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \end{array} \]

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 (- (* x y) (* (* z 9.0) t))))
   (if (<= t_1 (- INFINITY))
     (/ 1.0 (/ (/ a x) (* y 0.5)))
     (if (<= t_1 4e+304)
       (/ (fma x y (* z (* t -9.0))) (* a 2.0))
       (* t (* z (/ -4.5 a)))))))

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) - ((z * 9.0) * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 1.0 / ((a / x) / (y * 0.5));
	} else if (t_1 <= 4e+304) {
		tmp = fma(x, y, (z * (t * -9.0))) / (a * 2.0);
	} else {
		tmp = t * (z * (-4.5 / a));
	}
	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 * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(1.0 / Float64(Float64(a / x) / Float64(y * 0.5)));
	elseif (t_1 <= 4e+304)
		tmp = Float64(fma(x, y, Float64(z * Float64(t * -9.0))) / Float64(a * 2.0));
	else
		tmp = Float64(t * Float64(z * Float64(-4.5 / a)));
	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 * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(1.0 / N[(N[(a / x), $MachinePrecision] / N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+304], N[(N[(x * y + N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(z * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.5
Target	5.7
Herbie	4.2

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

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 63.8

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}} \]
      Proof
      (/.f64 (fma.f64 x y (*.f64 z (*.f64 t -9))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 x y (*.f64 z (*.f64 t (Rewrite<= metadata-eval (neg.f64 9))))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 x y (*.f64 z (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 t 9))))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 x y (*.f64 z (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 9 t))))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 x y (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 z (*.f64 9 t))))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 x y (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z 9) t)))) (*.f64 a 2)): 19 points increase in error, 6 points decrease in error
      (/.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))) (*.f64 a 2)): 0 points increase in error, 1 points decrease in error
      (/.f64 (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 x y) 0)) (*.f64 (*.f64 z 9) t)) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (+.f64 (*.f64 x y) 0) (*.f64 a 2)) (/.f64 (*.f64 (*.f64 z 9) t) (*.f64 a 2)))): 3 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite=> +-rgt-identity_binary64 (*.f64 x y)) (*.f64 a 2)) (/.f64 (*.f64 (*.f64 z 9) t) (*.f64 a 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) (*.f64 a 2))): 0 points increase in error, 3 points decrease in error

   3. Taylor expanded in x around inf 63.1

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot x}{a}} \]

   4. Simplified 31.4

      \[\leadsto \color{blue}{\frac{0.5 \cdot y}{\frac{a}{x}}} \]
      Proof
      (/.f64 (*.f64 1/2 y) (/.f64 a x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 y (/.f64 a x)))): 1 points increase in error, 0 points decrease in error
      (*.f64 1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y x) a))): 46 points increase in error, 50 points decrease in error

   5. Applied egg-rr 31.4

      \[\leadsto \color{blue}{\frac{y}{a} \cdot \frac{0.5}{\frac{1}{x}}} \]

   6. Applied egg-rr 31.4

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{x}}{y \cdot 0.5}}} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 3.9999999999999998e304

   1. Initial program 0.9

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

   2. Simplified 0.9

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}} \]
      Proof
      (/.f64 (fma.f64 x y (*.f64 z (*.f64 t -9))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 x y (*.f64 z (*.f64 t (Rewrite<= metadata-eval (neg.f64 9))))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 x y (*.f64 z (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 t 9))))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 x y (*.f64 z (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 9 t))))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 x y (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 z (*.f64 9 t))))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 x y (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z 9) t)))) (*.f64 a 2)): 19 points increase in error, 6 points decrease in error
      (/.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))) (*.f64 a 2)): 0 points increase in error, 1 points decrease in error
      (/.f64 (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 x y) 0)) (*.f64 (*.f64 z 9) t)) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (+.f64 (*.f64 x y) 0) (*.f64 a 2)) (/.f64 (*.f64 (*.f64 z 9) t) (*.f64 a 2)))): 3 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite=> +-rgt-identity_binary64 (*.f64 x y)) (*.f64 a 2)) (/.f64 (*.f64 (*.f64 z 9) t) (*.f64 a 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) (*.f64 a 2))): 0 points increase in error, 3 points decrease in error

   if 3.9999999999999998e304 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 61.5

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

   2. Taylor expanded in x around 0 61.9

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

   3. Simplified 62.0

      \[\leadsto \frac{\color{blue}{t \cdot \left(z \cdot -9\right)}}{a \cdot 2} \]
      Proof
      (*.f64 t (*.f64 z -9)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 t z) -9)): 42 points increase in error, 36 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 z t)) -9): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -9 (*.f64 z t))): 0 points increase in error, 0 points decrease in error
      (*.f64 -9 (Rewrite=> *-commutative_binary64 (*.f64 t z))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around 0 61.6

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

   5. Simplified 32.1

      \[\leadsto \color{blue}{t \cdot \left(z \cdot \frac{-4.5}{a}\right)} \]
      Proof
      (*.f64 t (*.f64 z (/.f64 -9/2 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 t (*.f64 z (/.f64 (Rewrite<= metadata-eval (/.f64 -9 2)) a))): 0 points increase in error, 0 points decrease in error
      (*.f64 t (*.f64 z (Rewrite<= associate-/r*_binary64 (/.f64 -9 (*.f64 2 a))))): 0 points increase in error, 0 points decrease in error
      (*.f64 t (*.f64 z (/.f64 -9 (Rewrite<= count-2_binary64 (+.f64 a a))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t z) (/.f64 -9 (+.f64 a a)))): 46 points increase in error, 46 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (*.f64 t z) -9) (+.f64 a a))): 30 points increase in error, 28 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -9 (*.f64 t z))) (+.f64 a a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -9 (*.f64 t z)) (Rewrite=> count-2_binary64 (*.f64 2 a))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> times-frac_binary64 (*.f64 (/.f64 -9 2) (/.f64 (*.f64 t z) a))): 32 points increase in error, 42 points decrease in error
      (*.f64 (Rewrite=> metadata-eval -9/2) (/.f64 (*.f64 t z) a)): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 4.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{x}}{y \cdot 0.5}}\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	4.2
Cost	2120

\[\begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{x}}{y \cdot 0.5}}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\frac{t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(z \cdot \frac{-4.5}{a}\right)\\ \end{array} \]

Alternative 2
Error	5.2
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+203}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\frac{0.5}{\frac{a}{x \cdot y + z \cdot \left(t \cdot -9\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 3
Error	25.1
Cost	1240

\[\begin{array}{l} t_1 := -4.5 \cdot \frac{z \cdot t}{a}\\ t_2 := 0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+74}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -920000000000:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-148}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	25.1
Cost	1240

\[\begin{array}{l} t_1 := 0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -100000000000:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{-93}:\\ \;\;\;\;\frac{-4.5}{\frac{a}{z \cdot t}}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-148}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-151}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	25.0
Cost	1240

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+74}:\\ \;\;\;\;\frac{y \cdot 0.5}{\frac{a}{x}}\\ \mathbf{elif}\;x \leq -1080000000000:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{-4.5}{\frac{a}{z \cdot t}}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-148}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-151}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \end{array} \]

Alternative 6
Error	25.0
Cost	1240

\[\begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+74}:\\ \;\;\;\;\frac{y \cdot 0.5}{\frac{a}{x}}\\ \mathbf{elif}\;x \leq -390000000000:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-93}:\\ \;\;\;\;\frac{-4.5}{\frac{a}{z \cdot t}}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-148}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-151}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \end{array} \]

Alternative 7
Error	23.8
Cost	976

\[\begin{array}{l} t_1 := 0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -480000000000:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-151}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	23.9
Cost	976

\[\begin{array}{l} t_1 := 0.5 \cdot \left(y \cdot \frac{x}{a}\right)\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -700000000000:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-151}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	32.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-249}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \end{array} \]

Alternative 10
Error	32.1
Cost	448

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

Reproduce

herbie shell --seed 2022331 
(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)))