Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2

Average Error: 24.8 → 7.0

Time: 8.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{z \cdot z - t \cdot a}\\ \mathbf{if}\;z \leq -1.62 \cdot 10^{+152}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-200}:\\ \;\;\;\;\frac{y}{\frac{\frac{t_1}{z}}{x}}\\ \mathbf{elif}\;z \leq 370000000:\\ \;\;\;\;z \cdot \frac{y \cdot x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (sqrt (- (* z z) (* t a)))))
   (if (<= z -1.62e+152)
     (* y (- x))
     (if (<= z -1e-200)
       (/ y (/ (/ t_1 z) x))
       (if (<= z 370000000.0) (* z (/ (* y x) t_1)) (* y x))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = sqrt(((z * z) - (t * a)));
	double tmp;
	if (z <= -1.62e+152) {
		tmp = y * -x;
	} else if (z <= -1e-200) {
		tmp = y / ((t_1 / z) / x);
	} else if (z <= 370000000.0) {
		tmp = z * ((y * x) / t_1);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = ((x * y) * z) / sqrt(((z * z) - (t * a)))
end function

↓

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((z * z) - (t * a)))
    if (z <= (-1.62d+152)) then
        tmp = y * -x
    else if (z <= (-1d-200)) then
        tmp = y / ((t_1 / z) / x)
    else if (z <= 370000000.0d0) then
        tmp = z * ((y * x) / t_1)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.sqrt(((z * z) - (t * a)));
	double tmp;
	if (z <= -1.62e+152) {
		tmp = y * -x;
	} else if (z <= -1e-200) {
		tmp = y / ((t_1 / z) / x);
	} else if (z <= 370000000.0) {
		tmp = z * ((y * x) / t_1);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	t_1 = math.sqrt(((z * z) - (t * a)))
	tmp = 0
	if z <= -1.62e+152:
		tmp = y * -x
	elif z <= -1e-200:
		tmp = y / ((t_1 / z) / x)
	elif z <= 370000000.0:
		tmp = z * ((y * x) / t_1)
	else:
		tmp = y * x
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = sqrt(Float64(Float64(z * z) - Float64(t * a)))
	tmp = 0.0
	if (z <= -1.62e+152)
		tmp = Float64(y * Float64(-x));
	elseif (z <= -1e-200)
		tmp = Float64(y / Float64(Float64(t_1 / z) / x));
	elseif (z <= 370000000.0)
		tmp = Float64(z * Float64(Float64(y * x) / t_1));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = sqrt(((z * z) - (t * a)));
	tmp = 0.0;
	if (z <= -1.62e+152)
		tmp = y * -x;
	elseif (z <= -1e-200)
		tmp = y / ((t_1 / z) / x);
	elseif (z <= 370000000.0)
		tmp = z * ((y * x) / t_1);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.62e+152], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, -1e-200], N[(y / N[(N[(t$95$1 / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 370000000.0], N[(z * N[(N[(y * x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.8
Target	7.9
Herbie	7.0

\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if z < -1.6200000000000001e152

   1. Initial program 53.3

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

   2. Taylor expanded in z around -inf 1.3

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot x\right)} \]

   3. Simplified 1.3

      \[\leadsto \color{blue}{y \cdot \left(-x\right)} \]

   if -1.6200000000000001e152 < z < -9.9999999999999998e-201

   1. Initial program 9.8

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

   2. Applied egg-rr 12.1

      \[\leadsto \color{blue}{\frac{1}{{\left(z \cdot z - t \cdot a\right)}^{0.25}} \cdot \frac{y \cdot \left(x \cdot z\right)}{{\left(z \cdot z - t \cdot a\right)}^{0.25}}} \]

   3. Applied egg-rr 9.9

      \[\leadsto \color{blue}{\frac{y}{{\left(z \cdot z - t \cdot a\right)}^{0.25} \cdot \frac{{\left(z \cdot z - t \cdot a\right)}^{0.25}}{z \cdot x}}} \]

   4. Applied egg-rr 6.3

      \[\leadsto \frac{y}{\color{blue}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{x}}} \]

   if -9.9999999999999998e-201 < z < 3.7e8

   1. Initial program 13.8

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

   2. Applied egg-rr 13.5

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

   if 3.7e8 < z

   1. Initial program 34.1

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

   2. Taylor expanded in z around inf 4.7

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 7.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.62 \cdot 10^{+152}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-200}:\\ \;\;\;\;\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{x}}\\ \mathbf{elif}\;z \leq 370000000:\\ \;\;\;\;z \cdot \frac{y \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Reproduce

herbie shell --seed 2022172 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z -3.1921305903852764e+46) (- (* y x)) (if (< z 5.976268120920894e+90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))