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

Average Error: 24.7 → 6.3

Time: 29.9s

Precision: binary64

Cost: 7496

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+154}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 10^{+82}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

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
 (if (<= z -1e+154)
   (* y (- x))
   (if (<= z 1e+82) (* x (* y (/ z (sqrt (- (* z z) (* t a)))))) (* 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 tmp;
	if (z <= -1e+154) {
		tmp = y * -x;
	} else if (z <= 1e+82) {
		tmp = x * (y * (z / sqrt(((z * z) - (t * a)))));
	} 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) :: tmp
    if (z <= (-1d+154)) then
        tmp = y * -x
    else if (z <= 1d+82) then
        tmp = x * (y * (z / sqrt(((z * z) - (t * a)))))
    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 tmp;
	if (z <= -1e+154) {
		tmp = y * -x;
	} else if (z <= 1e+82) {
		tmp = x * (y * (z / Math.sqrt(((z * z) - (t * a)))));
	} 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):
	tmp = 0
	if z <= -1e+154:
		tmp = y * -x
	elif z <= 1e+82:
		tmp = x * (y * (z / math.sqrt(((z * z) - (t * a)))))
	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)
	tmp = 0.0
	if (z <= -1e+154)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1e+82)
		tmp = Float64(x * Float64(y * Float64(z / sqrt(Float64(Float64(z * z) - Float64(t * a))))));
	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)
	tmp = 0.0;
	if (z <= -1e+154)
		tmp = y * -x;
	elseif (z <= 1e+82)
		tmp = x * (y * (z / sqrt(((z * z) - (t * a)))));
	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_] := If[LessEqual[z, -1e+154], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1e+82], N[(x * N[(y * N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Target

Original	24.7
Target	7.8
Herbie	6.3

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

Derivation

1. Split input into 3 regimes

2. if z < -1.00000000000000004e154

   1. Initial program 53.5

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

   2. Simplified 53.1

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

      [Start] 53.5	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 53.1	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]
      associate-*l* [=>] 53.1	\[ \color{blue}{x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)} \]

   3. Taylor expanded in z around -inf 1.5

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

   4. Simplified 1.5

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

      [Start] 1.5	\[ -1 \cdot \left(y \cdot x\right) \]
      associate-*r* [=>] 1.5	\[ \color{blue}{\left(-1 \cdot y\right) \cdot x} \]
      mul-1-neg [=>] 1.5	\[ \color{blue}{\left(-y\right)} \cdot x \]

   if -1.00000000000000004e154 < z < 9.9999999999999996e81

   1. Initial program 11.2

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

   2. Simplified 8.6

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

      [Start] 11.2	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 9.0	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]
      associate-*l* [=>] 8.6	\[ \color{blue}{x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)} \]

   if 9.9999999999999996e81 < z

   1. Initial program 42.0

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

   2. Simplified 39.3

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

      [Start] 42.0	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 39.3	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]
      associate-*l* [=>] 39.3	\[ \color{blue}{x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)} \]

   3. Taylor expanded in z around inf 3.1

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

4. Final simplification 6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+154}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 10^{+82}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Error	12.0
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-144}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\left(a \cdot 0.5\right) \cdot \frac{t}{z} - z}{z}}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-131}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	11.9
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-144}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\left(a \cdot 0.5\right) \cdot \frac{t}{z} - z}{z}}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-132}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{a \cdot \left(-t\right)}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Error	16.4
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-139}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-132}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4
Error	16.1
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-148}:\\ \;\;\;\;z \cdot \frac{y}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5
Error	15.8
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+66}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Error	16.1
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-56}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{\frac{\frac{\left(a \cdot \frac{t}{z}\right) \cdot -0.5 - z}{y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7
Error	15.8
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+73}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-152}:\\ \;\;\;\;\frac{y}{\frac{0.5 \cdot \left(a \cdot \frac{t}{z}\right) - z}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8
Error	16.8
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-135}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.42 \cdot 10^{-148}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(\frac{z}{t} \cdot \left(-2 \cdot \frac{y}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 9
Error	15.5
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq 7.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\left(a \cdot 0.5\right) \cdot \frac{t}{z} - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10
Error	16.0
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{-154}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \frac{t \cdot a}{z} - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11
Error	18.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-161}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{x}{\frac{z}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 12
Error	17.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-139}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-150}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 13
Error	17.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-61}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-150}:\\ \;\;\;\;-1 + \left(1 - y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 14
Error	19.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 15
Error	36.4
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023060 
(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)))))