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

Average Error: 24.8 → 6.9

Time: 33.4s

Precision: binary64

Cost: 13960

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+154}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-254}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{1 - \frac{a}{z} \cdot \frac{t}{z}}}{y \cdot x}\right)}^{-1}\\ \end{array} \]

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
 (if (<= z -4.9e+154)
   (* y (- x))
   (if (<= z 1.65e-254)
     (* x (* y (/ z (sqrt (- (* z z) (* t a))))))
     (pow (/ (sqrt (- 1.0 (* (/ a z) (/ t z)))) (* y x)) -1.0))))

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 tmp;
	if (z <= -4.9e+154) {
		tmp = y * -x;
	} else if (z <= 1.65e-254) {
		tmp = x * (y * (z / sqrt(((z * z) - (t * a)))));
	} else {
		tmp = pow((sqrt((1.0 - ((a / z) * (t / z)))) / (y * x)), -1.0);
	}
	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) :: tmp
    if (z <= (-4.9d+154)) then
        tmp = y * -x
    else if (z <= 1.65d-254) then
        tmp = x * (y * (z / sqrt(((z * z) - (t * a)))))
    else
        tmp = (sqrt((1.0d0 - ((a / z) * (t / z)))) / (y * x)) ** (-1.0d0)
    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 tmp;
	if (z <= -4.9e+154) {
		tmp = y * -x;
	} else if (z <= 1.65e-254) {
		tmp = x * (y * (z / Math.sqrt(((z * z) - (t * a)))));
	} else {
		tmp = Math.pow((Math.sqrt((1.0 - ((a / z) * (t / z)))) / (y * x)), -1.0);
	}
	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):
	tmp = 0
	if z <= -4.9e+154:
		tmp = y * -x
	elif z <= 1.65e-254:
		tmp = x * (y * (z / math.sqrt(((z * z) - (t * a)))))
	else:
		tmp = math.pow((math.sqrt((1.0 - ((a / z) * (t / z)))) / (y * x)), -1.0)
	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)
	tmp = 0.0
	if (z <= -4.9e+154)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.65e-254)
		tmp = Float64(x * Float64(y * Float64(z / sqrt(Float64(Float64(z * z) - Float64(t * a))))));
	else
		tmp = Float64(sqrt(Float64(1.0 - Float64(Float64(a / z) * Float64(t / z)))) / Float64(y * x)) ^ -1.0;
	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)
	tmp = 0.0;
	if (z <= -4.9e+154)
		tmp = y * -x;
	elseif (z <= 1.65e-254)
		tmp = x * (y * (z / sqrt(((z * z) - (t * a)))));
	else
		tmp = (sqrt((1.0 - ((a / z) * (t / z)))) / (y * x)) ^ -1.0;
	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_] := If[LessEqual[z, -4.9e+154], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.65e-254], N[(x * N[(y * N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[N[(1.0 - N[(N[(a / z), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]

Target

Original	24.8
Target	7.6
Herbie	6.9

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

Derivation

1. Split input into 3 regimes

2. if z < -4.9000000000000002e154

   1. Initial program 53.7

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

   2. Simplified 53.3

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

      [Start] 53.7	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 53.3	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]
      associate-*l* [=>] 53.3	\[ \color{blue}{x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)} \]

   3. Taylor expanded in z around -inf 1.4

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

   4. Simplified 1.4

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

      [Start] 1.4	\[ -1 \cdot \left(y \cdot x\right) \]
      associate-*r* [=>] 1.4	\[ \color{blue}{\left(-1 \cdot y\right) \cdot x} \]
      mul-1-neg [=>] 1.4	\[ \color{blue}{\left(-y\right)} \cdot x \]

   if -4.9000000000000002e154 < z < 1.65000000000000008e-254

   1. Initial program 11.9

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

   2. Simplified 9.4

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

      [Start] 11.9	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 9.5	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]
      associate-*l* [=>] 9.4	\[ \color{blue}{x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)} \]

   if 1.65000000000000008e-254 < z

   1. Initial program 25.5

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

   2. Simplified 23.4

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

      [Start] 25.5	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-/l* [=>] 23.4	\[ \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Applied egg-rr 30.3

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

   4. Simplified 9.0

      \[\leadsto \frac{x \cdot y}{\color{blue}{\sqrt{1 - \frac{a}{z \cdot z} \cdot t}}} \]
      Proof

      [Start] 30.3	\[ \frac{x \cdot y}{\sqrt{\frac{z \cdot z - t \cdot a}{z \cdot z}}} \]
      *-commutative [=>] 30.3	\[ \frac{x \cdot y}{\sqrt{\frac{z \cdot z - \color{blue}{a \cdot t}}{z \cdot z}}} \]
      div-sub [=>] 33.2	\[ \frac{x \cdot y}{\sqrt{\color{blue}{\frac{z \cdot z}{z \cdot z} - \frac{a \cdot t}{z \cdot z}}}} \]
      unpow2 [<=] 33.2	\[ \frac{x \cdot y}{\sqrt{\frac{\color{blue}{{z}^{2}}}{z \cdot z} - \frac{a \cdot t}{z \cdot z}}} \]
      unpow2 [<=] 33.2	\[ \frac{x \cdot y}{\sqrt{\frac{{z}^{2}}{\color{blue}{{z}^{2}}} - \frac{a \cdot t}{z \cdot z}}} \]
      *-inverses [=>] 11.2	\[ \frac{x \cdot y}{\sqrt{\color{blue}{1} - \frac{a \cdot t}{z \cdot z}}} \]
      unpow2 [<=] 11.2	\[ \frac{x \cdot y}{\sqrt{1 - \frac{a \cdot t}{\color{blue}{{z}^{2}}}}} \]
      associate-/l* [=>] 8.9	\[ \frac{x \cdot y}{\sqrt{1 - \color{blue}{\frac{a}{\frac{{z}^{2}}{t}}}}} \]
      associate-/r/ [=>] 9.0	\[ \frac{x \cdot y}{\sqrt{1 - \color{blue}{\frac{a}{{z}^{2}} \cdot t}}} \]
      unpow2 [=>] 9.0	\[ \frac{x \cdot y}{\sqrt{1 - \frac{a}{\color{blue}{z \cdot z}} \cdot t}} \]

   5. Applied egg-rr 6.7

      \[\leadsto \frac{x \cdot y}{\sqrt{1 - \color{blue}{\frac{\frac{a}{z} \cdot t}{z}}}} \]

   6. Applied egg-rr 6.9

      \[\leadsto \color{blue}{{\left(\frac{\sqrt{1 - \frac{a}{z} \cdot \frac{t}{z}}}{x \cdot y}\right)}^{-1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+154}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-254}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{1 - \frac{a}{z} \cdot \frac{t}{z}}}{y \cdot x}\right)}^{-1}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.2
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+154}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 10^{+134}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 2
Error	6.8
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+154}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-272}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{1 - \frac{a}{z} \cdot \frac{t}{z}}}\\ \end{array} \]

Alternative 3
Error	14.4
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+38}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-201}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 4
Error	14.2
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+38}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-200}:\\ \;\;\;\;z \cdot \frac{x}{\frac{\sqrt{t \cdot \left(-a\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 5
Error	15.1
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+22}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-282}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{\left(t \cdot a\right) \cdot \frac{0.5}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 6
Error	14.6
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+211}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-280}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{0.5 \cdot \frac{t \cdot a}{z} - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 7
Error	17.5
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-119}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-98}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8
Error	17.6
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-116}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-99}:\\ \;\;\;\;z \cdot \left(\frac{z \cdot x}{\frac{a}{2}} \cdot \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 9
Error	17.4
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-116}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.04 \cdot 10^{-100}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{0.5 \cdot \frac{t}{\frac{z}{a}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10
Error	17.5
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{-117}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{\frac{t}{z} \cdot \left(a \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11
Error	15.9
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-118}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\right)\\ \end{array} \]

Alternative 12
Error	15.9
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-118}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 13
Error	17.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-118}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 14
Error	19.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -8.6 \cdot 10^{-119}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-125}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 15
Error	18.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-119}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-125}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 16
Error	19.3
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-282}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 17
Error	37.0
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023016 
(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)))))