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

Average Error: 21.3 → 17.2

Time: 15.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \cos \left(y - \frac{z \cdot t}{3}\right)\\ \mathbf{if}\;t_2 \leq 0.9999999999999992:\\ \;\;\;\;t_1 \cdot t_2 - 0.3333333333333333 \cdot \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1 - \frac{a}{b \cdot 3}\\ \end{array} \]

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 (cos (- y (/ (* z t) 3.0)))))
   (if (<= t_2 0.9999999999999992)
     (- (* t_1 t_2) (* 0.3333333333333333 (/ a b)))
     (- t_1 (/ a (* b 3.0))))))

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 = cos((y - ((z * t) / 3.0)));
	double tmp;
	if (t_2 <= 0.9999999999999992) {
		tmp = (t_1 * t_2) - (0.3333333333333333 * (a / b));
	} else {
		tmp = t_1 - (a / (b * 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((2.0d0 * sqrt(x)) * cos((y - ((z * t) / 3.0d0)))) - (a / (b * 3.0d0))
end function

↓

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 * sqrt(x)
    t_2 = cos((y - ((z * t) / 3.0d0)))
    if (t_2 <= 0.9999999999999992d0) then
        tmp = (t_1 * t_2) - (0.3333333333333333d0 * (a / b))
    else
        tmp = t_1 - (a / (b * 3.0d0))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 2.0 * Math.sqrt(x);
	double t_2 = Math.cos((y - ((z * t) / 3.0)));
	double tmp;
	if (t_2 <= 0.9999999999999992) {
		tmp = (t_1 * t_2) - (0.3333333333333333 * (a / b));
	} else {
		tmp = t_1 - (a / (b * 3.0));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	return ((2.0 * math.sqrt(x)) * math.cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0))

↓

def code(x, y, z, t, a, b):
	t_1 = 2.0 * math.sqrt(x)
	t_2 = math.cos((y - ((z * t) / 3.0)))
	tmp = 0
	if t_2 <= 0.9999999999999992:
		tmp = (t_1 * t_2) - (0.3333333333333333 * (a / b))
	else:
		tmp = t_1 - (a / (b * 3.0))
	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 = cos(Float64(y - Float64(Float64(z * t) / 3.0)))
	tmp = 0.0
	if (t_2 <= 0.9999999999999992)
		tmp = Float64(Float64(t_1 * t_2) - Float64(0.3333333333333333 * Float64(a / b)));
	else
		tmp = Float64(t_1 - Float64(a / Float64(b * 3.0)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
end

↓

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 2.0 * sqrt(x);
	t_2 = cos((y - ((z * t) / 3.0)));
	tmp = 0.0;
	if (t_2 <= 0.9999999999999992)
		tmp = (t_1 * t_2) - (0.3333333333333333 * (a / b));
	else
		tmp = t_1 - (a / (b * 3.0));
	end
	tmp_2 = 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[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.9999999999999992], N[(N[(t$95$1 * t$95$2), $MachinePrecision] - N[(0.3333333333333333 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $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	21.3
Target	19.2
Herbie	17.2

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))) < 0.999999999999999223

   1. Initial program 20.2

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

   2. Taylor expanded in a around 0 20.3

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

   if 0.999999999999999223 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))

   1. Initial program 23.1

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

   2. Taylor expanded in z around 0 11.8

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

   3. Taylor expanded in y around 0 11.8

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

4. Final simplification 17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.9999999999999992:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - 0.3333333333333333 \cdot \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{b \cdot 3}\\ \end{array} \]

Reproduce

herbie shell --seed 2022152 
(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))))