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

Average Accuracy: 69.7% → 79.2%

Time: 37.4s

Precision: binary64

Cost: 48840

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

↓

\[\begin{array}{l} t_1 := \left(2 \cdot \sqrt{x}\right) \cdot \cos y\\ t_2 := z \cdot \left(t \cdot 0.3333333333333333\right)\\ t_3 := y - t_2\\ t_4 := \mathsf{fma}\left(-t, z \cdot 0.3333333333333333, t_2\right)\\ t_5 := y - \frac{z \cdot t}{3}\\ t_6 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t_5 \leq -\infty:\\ \;\;\;\;t_1 - t_6\\ \mathbf{elif}\;t_5 \leq 2 \cdot 10^{+274}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos t_3 \cdot \cos t_4 - \sin t_3 \cdot \sin t_4\right)\right) - t_6\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{a}{3} \cdot \frac{-1}{b}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (* 2.0 (sqrt x)) (cos y)))
        (t_2 (* z (* t 0.3333333333333333)))
        (t_3 (- y t_2))
        (t_4 (fma (- t) (* z 0.3333333333333333) t_2))
        (t_5 (- y (/ (* z t) 3.0)))
        (t_6 (/ a (* 3.0 b))))
   (if (<= t_5 (- INFINITY))
     (- t_1 t_6)
     (if (<= t_5 2e+274)
       (-
        (*
         2.0
         (* (sqrt x) (- (* (cos t_3) (cos t_4)) (* (sin t_3) (sin t_4)))))
        t_6)
       (+ t_1 (* (/ a 3.0) (/ -1.0 b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (2.0 * sqrt(x)) * cos(y);
	double t_2 = z * (t * 0.3333333333333333);
	double t_3 = y - t_2;
	double t_4 = fma(-t, (z * 0.3333333333333333), t_2);
	double t_5 = y - ((z * t) / 3.0);
	double t_6 = a / (3.0 * b);
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = t_1 - t_6;
	} else if (t_5 <= 2e+274) {
		tmp = (2.0 * (sqrt(x) * ((cos(t_3) * cos(t_4)) - (sin(t_3) * sin(t_4))))) - t_6;
	} else {
		tmp = t_1 + ((a / 3.0) * (-1.0 / b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end

↓

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(2.0 * sqrt(x)) * cos(y))
	t_2 = Float64(z * Float64(t * 0.3333333333333333))
	t_3 = Float64(y - t_2)
	t_4 = fma(Float64(-t), Float64(z * 0.3333333333333333), t_2)
	t_5 = Float64(y - Float64(Float64(z * t) / 3.0))
	t_6 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(t_1 - t_6);
	elseif (t_5 <= 2e+274)
		tmp = Float64(Float64(2.0 * Float64(sqrt(x) * Float64(Float64(cos(t_3) * cos(t_4)) - Float64(sin(t_3) * sin(t_4))))) - t_6);
	else
		tmp = Float64(t_1 + Float64(Float64(a / 3.0) * Float64(-1.0 / b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(t * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y - t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[((-t) * N[(z * 0.3333333333333333), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(t$95$1 - t$95$6), $MachinePrecision], If[LessEqual[t$95$5, 2e+274], N[(N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(N[(N[Cos[t$95$3], $MachinePrecision] * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$3], $MachinePrecision] * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision], N[(t$95$1 + N[(N[(a / 3.0), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Target

Original	69.7%
Target	73.6%
Herbie	79.2%

\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (-.f64 y (/.f64 (*.f64 z t) 3)) < -inf.0

   1. Initial program 0.0%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

   2. Taylor expanded in z around 0 49.4%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

   if -inf.0 < (-.f64 y (/.f64 (*.f64 z t) 3)) < 1.99999999999999984e274

   1. Initial program 81.1%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

   2. Simplified 81.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z}{\frac{3}{t}}\right)\right) - \frac{a}{3 \cdot b}} \]
      Proof

      [Start] 81.1	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
      associate-*l* [=>] 81.1	\[ \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3} \]
      fma-neg [=>] 81.1	\[ \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right), -\frac{a}{b \cdot 3}\right)} \]
      remove-double-neg [<=] 81.1	\[ \mathsf{fma}\left(2, \color{blue}{-\left(-\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)}, -\frac{a}{b \cdot 3}\right) \]
      fma-neg [<=] 81.1	\[ \color{blue}{2 \cdot \left(-\left(-\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}} \]
      remove-double-neg [=>] 81.1	\[ 2 \cdot \color{blue}{\left(\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3} \]
      associate-/l* [=>] 81.1	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \color{blue}{\frac{z}{\frac{3}{t}}}\right)\right) - \frac{a}{b \cdot 3} \]
      *-commutative [=>] 81.1	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z}{\frac{3}{t}}\right)\right) - \frac{a}{\color{blue}{3 \cdot b}} \]

   3. Applied egg-rr 85.9%

      \[\leadsto 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(\cos \left(y - z \cdot \left(t \cdot 0.3333333333333333\right)\right) \cdot \cos \left(\mathsf{fma}\left(-t, z \cdot 0.3333333333333333, z \cdot \left(t \cdot 0.3333333333333333\right)\right)\right) - \sin \left(y - z \cdot \left(t \cdot 0.3333333333333333\right)\right) \cdot \sin \left(\mathsf{fma}\left(-t, z \cdot 0.3333333333333333, z \cdot \left(t \cdot 0.3333333333333333\right)\right)\right)\right)}\right) - \frac{a}{3 \cdot b} \]
      Proof

      [Start] 81.1	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z}{\frac{3}{t}}\right)\right) - \frac{a}{3 \cdot b} \]
      *-un-lft-identity [=>] 81.1	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(\color{blue}{1 \cdot y} - \frac{z}{\frac{3}{t}}\right)\right) - \frac{a}{3 \cdot b} \]
      associate-/r/ [=>] 81.4	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(1 \cdot y - \color{blue}{\frac{z}{3} \cdot t}\right)\right) - \frac{a}{3 \cdot b} \]
      prod-diff [=>] 81.4	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \color{blue}{\left(\mathsf{fma}\left(1, y, -t \cdot \frac{z}{3}\right) + \mathsf{fma}\left(-t, \frac{z}{3}, t \cdot \frac{z}{3}\right)\right)}\right) - \frac{a}{3 \cdot b} \]
      *-commutative [<=] 81.4	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(\mathsf{fma}\left(1, y, -\color{blue}{\frac{z}{3} \cdot t}\right) + \mathsf{fma}\left(-t, \frac{z}{3}, t \cdot \frac{z}{3}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      associate-/r/ [<=] 81.1	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(\mathsf{fma}\left(1, y, -\color{blue}{\frac{z}{\frac{3}{t}}}\right) + \mathsf{fma}\left(-t, \frac{z}{3}, t \cdot \frac{z}{3}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      fma-neg [<=] 81.1	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(\color{blue}{\left(1 \cdot y - \frac{z}{\frac{3}{t}}\right)} + \mathsf{fma}\left(-t, \frac{z}{3}, t \cdot \frac{z}{3}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      *-un-lft-identity [<=] 81.1	\[ 2 \cdot \left(\sqrt{x} \cdot \cos \left(\left(\color{blue}{y} - \frac{z}{\frac{3}{t}}\right) + \mathsf{fma}\left(-t, \frac{z}{3}, t \cdot \frac{z}{3}\right)\right)\right) - \frac{a}{3 \cdot b} \]
      cos-sum [=>] 84.4	\[ 2 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(\cos \left(y - \frac{z}{\frac{3}{t}}\right) \cdot \cos \left(\mathsf{fma}\left(-t, \frac{z}{3}, t \cdot \frac{z}{3}\right)\right) - \sin \left(y - \frac{z}{\frac{3}{t}}\right) \cdot \sin \left(\mathsf{fma}\left(-t, \frac{z}{3}, t \cdot \frac{z}{3}\right)\right)\right)}\right) - \frac{a}{3 \cdot b} \]

   if 1.99999999999999984e274 < (-.f64 y (/.f64 (*.f64 z t) 3))

   1. Initial program 19.5%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

   2. Taylor expanded in z around 0 55.9%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]

   3. Applied egg-rr 56.3%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{1}{b} \cdot \frac{a}{3}} \]
      Proof

      [Start] 55.9	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{b \cdot 3} \]
      *-un-lft-identity [=>] 55.9	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\color{blue}{1 \cdot a}}{b \cdot 3} \]
      times-frac [=>] 56.3	\[ \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{1}{b} \cdot \frac{a}{3}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y - \frac{z \cdot t}{3} \leq -\infty:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;y - \frac{z \cdot t}{3} \leq 2 \cdot 10^{+274}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos \left(y - z \cdot \left(t \cdot 0.3333333333333333\right)\right) \cdot \cos \left(\mathsf{fma}\left(-t, z \cdot 0.3333333333333333, z \cdot \left(t \cdot 0.3333333333333333\right)\right)\right) - \sin \left(y - z \cdot \left(t \cdot 0.3333333333333333\right)\right) \cdot \sin \left(\mathsf{fma}\left(-t, z \cdot 0.3333333333333333, z \cdot \left(t \cdot 0.3333333333333333\right)\right)\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y + \frac{a}{3} \cdot \frac{-1}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	77.1%
Cost	46664

\[\begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+86}:\\ \;\;\;\;t_1 \cdot \cos y + \frac{a}{3} \cdot \frac{-1}{b}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x} \cdot \mathsf{fma}\left(\cos \left(t \cdot \left(z \cdot -0.3333333333333333\right)\right), \cos y, \sin y \cdot \sin \left(\left(z \cdot t\right) \cdot 0.3333333333333333\right)\right), -0.3333333333333333 \cdot \frac{a}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log t_1} - \frac{a}{3 \cdot b}\\ \end{array} \]

Alternative 2
Accuracy	71.6%
Cost	13896

\[\begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-48}:\\ \;\;\;\;t_2 - t_1\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{x} \cdot \left(2 \cdot \cos y\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 - \frac{\frac{a}{b}}{3}\\ \end{array} \]

Alternative 3
Accuracy	75.9%
Cost	13504

\[\left(2 \cdot \sqrt{x}\right) \cdot \cos y - a \cdot \frac{0.3333333333333333}{b} \]

Alternative 4
Accuracy	76.0%
Cost	13504

\[\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{3 \cdot b} \]

Alternative 5
Accuracy	76.0%
Cost	13504

\[\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\frac{a}{b}}{3} \]

Alternative 6
Accuracy	65.2%
Cost	6976

\[2 \cdot \sqrt{x} - 0.3333333333333333 \cdot \frac{a}{b} \]

Alternative 7
Accuracy	65.3%
Cost	6976

\[2 \cdot \sqrt{x} - \frac{a}{3 \cdot b} \]

Alternative 8
Accuracy	65.2%
Cost	6976

\[2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3} \]

Alternative 9
Accuracy	50.4%
Cost	320

\[-0.3333333333333333 \cdot \frac{a}{b} \]

Alternative 10
Accuracy	50.5%
Cost	320

\[\frac{a}{b \cdot -3} \]

Reproduce

herbie shell --seed 2023160 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1.0 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))