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

Average Error: 20.6 → 15.4

Time: 16.6s

Precision: binary64

\[[z, t] = \mathsf{sort}([z, t]) \\]

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := z \cdot \frac{t}{3}\\ t_3 := \mathsf{fma}\left(1, y, -t_2\right)\\ t_4 := y - \frac{z \cdot t}{3}\\ t_5 := \mathsf{fma}\left(-\frac{t}{3}, z, t_2\right)\\ t_6 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t_4 \leq -3.2697914759171068 \cdot 10^{+156}:\\ \;\;\;\;t_1 \cdot \cos y - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;t_4 \leq 1.0098144165049147 \cdot 10^{+297}:\\ \;\;\;\;t_1 \cdot \left(\cos t_3 \cdot \cos t_5 - \sin t_3 \cdot \sin t_5\right) - t_6\\ \mathbf{else}:\\ \;\;\;\;t_1 - t_6\\ \end{array} \]

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)))
        (t_2 (* z (/ t 3.0)))
        (t_3 (fma 1.0 y (- t_2)))
        (t_4 (- y (/ (* z t) 3.0)))
        (t_5 (fma (- (/ t 3.0)) z t_2))
        (t_6 (/ a (* 3.0 b))))
   (if (<= t_4 -3.2697914759171068e+156)
     (- (* t_1 (cos y)) (/ (/ a 3.0) b))
     (if (<= t_4 1.0098144165049147e+297)
       (- (* t_1 (- (* (cos t_3) (cos t_5)) (* (sin t_3) (sin t_5)))) t_6)
       (- t_1 t_6)))))

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);
	double t_2 = z * (t / 3.0);
	double t_3 = fma(1.0, y, -t_2);
	double t_4 = y - ((z * t) / 3.0);
	double t_5 = fma(-(t / 3.0), z, t_2);
	double t_6 = a / (3.0 * b);
	double tmp;
	if (t_4 <= -3.2697914759171068e+156) {
		tmp = (t_1 * cos(y)) - ((a / 3.0) / b);
	} else if (t_4 <= 1.0098144165049147e+297) {
		tmp = (t_1 * ((cos(t_3) * cos(t_5)) - (sin(t_3) * sin(t_5)))) - t_6;
	} else {
		tmp = t_1 - t_6;
	}
	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(2.0 * sqrt(x))
	t_2 = Float64(z * Float64(t / 3.0))
	t_3 = fma(1.0, y, Float64(-t_2))
	t_4 = Float64(y - Float64(Float64(z * t) / 3.0))
	t_5 = fma(Float64(-Float64(t / 3.0)), z, t_2)
	t_6 = Float64(a / Float64(3.0 * b))
	tmp = 0.0
	if (t_4 <= -3.2697914759171068e+156)
		tmp = Float64(Float64(t_1 * cos(y)) - Float64(Float64(a / 3.0) / b));
	elseif (t_4 <= 1.0098144165049147e+297)
		tmp = Float64(Float64(t_1 * Float64(Float64(cos(t_3) * cos(t_5)) - Float64(sin(t_3) * sin(t_5)))) - t_6);
	else
		tmp = Float64(t_1 - t_6);
	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[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(t / 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 * y + (-t$95$2)), $MachinePrecision]}, Block[{t$95$4 = N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[((-N[(t / 3.0), $MachinePrecision]) * z + t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[(a / N[(3.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -3.2697914759171068e+156], N[(N[(t$95$1 * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(N[(a / 3.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1.0098144165049147e+297], N[(N[(t$95$1 * N[(N[(N[Cos[t$95$3], $MachinePrecision] * N[Cos[t$95$5], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$3], $MachinePrecision] * N[Sin[t$95$5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision], N[(t$95$1 - t$95$6), $MachinePrecision]]]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	20.6
Target	18.7
Herbie	15.4

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 y (/.f64 (*.f64 z t) 3)) < -3.26979147591710678e156

   1. Initial program 33.0

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

   2. Taylor expanded in z around 0 24.2

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

   3. Applied *-un-lft-identity_binary64 24.2

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

   4. Applied times-frac_binary64 24.3

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

   5. Applied associate-*l/_binary64 24.2

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

   if -3.26979147591710678e156 < (-.f64 y (/.f64 (*.f64 z t) 3)) < 1.0098144165049147e297

   1. Initial program 12.7

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

   2. Applied *-un-lft-identity_binary64 12.7

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

   3. Applied times-frac_binary64 12.6

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

   4. Applied *-un-lft-identity_binary64 12.6

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

   5. Applied prod-diff_binary64 12.6

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right) + \mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)} - \frac{a}{b \cdot 3} \]

   6. Applied cos-sum_binary64 10.4

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{3} \cdot \frac{z}{1}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, \frac{z}{1}, \frac{t}{3} \cdot \frac{z}{1}\right)\right)\right)} - \frac{a}{b \cdot 3} \]

   if 1.0098144165049147e297 < (-.f64 y (/.f64 (*.f64 z t) 3))

   1. Initial program 57.2

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

   2. Taylor expanded in z around 0 32.3

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

   3. Taylor expanded in y around 0 34.4

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 15.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y - \frac{z \cdot t}{3} \leq -3.2697914759171068 \cdot 10^{+156}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;y - \frac{z \cdot t}{3} \leq 1.0098144165049147 \cdot 10^{+297}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))))