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

Average Accuracy: 94.4% → 99.2%

Time: 15.8s

Precision: binary64

Cost: 9289

Math

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

↓

\[\begin{array}{l} t_1 := \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{\frac{\frac{t}{z}}{y}}{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* y (* z 3.0))))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+306)))
     (+ x (fma -0.3333333333333333 (/ y z) (/ (/ (/ t z) y) 3.0)))
     t_1)))

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))) + (t / (y * (z * 3.0)));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+306)) {
		tmp = x + fma(-0.3333333333333333, (y / z), (((t / z) / y) / 3.0));
	} else {
		tmp = t_1;
	}
	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(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(y * Float64(z * 3.0))))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+306))
		tmp = Float64(x + fma(-0.3333333333333333, Float64(y / z), Float64(Float64(Float64(t / z) / y) / 3.0)));
	else
		tmp = t_1;
	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[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+306]], $MachinePrecision]], N[(x + N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + N[(N[(N[(t / z), $MachinePrecision] / y), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]

Target

Original	94.4%
Target	97.4%
Herbie	99.2%

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y))) < -inf.0 or 4.99999999999999993e306 < (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y)))

   1. Initial program 4.7%

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

   2. Simplified 98.8%

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

      [Start] 4.7	\[ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
      associate-+l- [=>] 4.7	\[ \color{blue}{x - \left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      sub-neg [=>] 4.7	\[ \color{blue}{x + \left(-\left(\frac{y}{z \cdot 3} - \frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)} \]
      sub-neg [=>] 4.7	\[ x + \left(-\color{blue}{\left(\frac{y}{z \cdot 3} + \left(-\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)}\right) \]
      distribute-neg-in [=>] 4.7	\[ x + \color{blue}{\left(\left(-\frac{y}{z \cdot 3}\right) + \left(-\left(-\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)\right)} \]
      distribute-neg-frac [=>] 4.7	\[ x + \left(\color{blue}{\frac{-y}{z \cdot 3}} + \left(-\left(-\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)\right) \]
      neg-mul-1 [=>] 4.7	\[ x + \left(\frac{\color{blue}{-1 \cdot y}}{z \cdot 3} + \left(-\left(-\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)\right) \]
      *-commutative [=>] 4.7	\[ x + \left(\frac{-1 \cdot y}{\color{blue}{3 \cdot z}} + \left(-\left(-\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)\right) \]
      times-frac [=>] 4.7	\[ x + \left(\color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} + \left(-\left(-\frac{t}{\left(z \cdot 3\right) \cdot y}\right)\right)\right) \]
      remove-double-neg [=>] 4.7	\[ x + \left(\frac{-1}{3} \cdot \frac{y}{z} + \color{blue}{\frac{t}{\left(z \cdot 3\right) \cdot y}}\right) \]
      fma-def [=>] 4.7	\[ x + \color{blue}{\mathsf{fma}\left(\frac{-1}{3}, \frac{y}{z}, \frac{t}{\left(z \cdot 3\right) \cdot y}\right)} \]
      metadata-eval [=>] 4.7	\[ x + \mathsf{fma}\left(\color{blue}{-0.3333333333333333}, \frac{y}{z}, \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]
      associate-*l* [=>] 4.7	\[ x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}}\right) \]
      associate-/r* [=>] 99.1	\[ x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \color{blue}{\frac{\frac{t}{z}}{3 \cdot y}}\right) \]
      associate-/l/ [<=] 98.8	\[ x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \color{blue}{\frac{\frac{\frac{t}{z}}{y}}{3}}\right) \]

   if -inf.0 < (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y))) < 4.99999999999999993e306

   1. Initial program 99.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq -\infty \lor \neg \left(\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{\frac{\frac{t}{z}}{y}}{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	3016

\[\begin{array}{l} t_1 := y - \frac{t}{y}\\ t_2 := \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+297}:\\ \;\;\;\;x + \frac{t_1}{z \cdot -3}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z} \cdot t_1\\ \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	3016

\[\begin{array}{l} t_1 := \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+304}:\\ \;\;\;\;\left(x - \frac{1}{z} \cdot \frac{y}{3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)\\ \end{array} \]

Alternative 3
Accuracy	55.8%
Cost	980

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.75 \cdot 10^{-174}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-279}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{-274}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	97.5%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-71} \lor \neg \left(y \leq 6 \cdot 10^{-117}\right):\\ \;\;\;\;x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \end{array} \]

Alternative 5
Accuracy	97.4%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-139} \lor \neg \left(y \leq 7.2 \cdot 10^{-127}\right):\\ \;\;\;\;x + \frac{-0.3333333333333333 \cdot \left(y - \frac{t}{y}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \end{array} \]

Alternative 6
Accuracy	97.4%
Cost	968

\[\begin{array}{l} t_1 := y - \frac{t}{y}\\ \mathbf{if}\;y \leq -2.45 \cdot 10^{-139}:\\ \;\;\;\;x + \frac{-0.3333333333333333 \cdot t_1}{z}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-125}:\\ \;\;\;\;x + \frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t_1}{z \cdot -3}\\ \end{array} \]

Alternative 7
Accuracy	55.2%
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-118}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-280}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-274}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	85.7%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-22} \lor \neg \left(y \leq 3900000000\right):\\ \;\;\;\;x - \frac{0.3333333333333333}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + 0.3333333333333333 \cdot \frac{\frac{t}{y}}{z}\\ \end{array} \]

Alternative 9
Accuracy	90.5%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-22} \lor \neg \left(y \leq 122000000\right):\\ \;\;\;\;x - \frac{0.3333333333333333}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \end{array} \]

Alternative 10
Accuracy	73.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{-180} \lor \neg \left(y \leq 3.8 \cdot 10^{-303}\right):\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \end{array} \]

Alternative 11
Accuracy	73.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-180}:\\ \;\;\;\;x - \frac{0.3333333333333333}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-301}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \end{array} \]

Alternative 12
Accuracy	73.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-180}:\\ \;\;\;\;x - \frac{0.3333333333333333}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-301}:\\ \;\;\;\;\frac{0.3333333333333333}{y \cdot \frac{z}{t}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \end{array} \]

Alternative 13
Accuracy	73.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-179}:\\ \;\;\;\;x - \frac{0.3333333333333333}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-301}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{y}{\frac{t}{z}}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \end{array} \]

Alternative 14
Accuracy	55.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-118}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.0025:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Accuracy	55.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-118}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Accuracy	55.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-118}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Accuracy	55.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-118}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 0.002:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 18
Accuracy	41.1%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023146 
(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))))