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

Average Error: 24.5 → 7.7

Time: 34.1s

Precision: binary64

Cost: 7496

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -620000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

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 -620000.0)
   (* y (- x))
   (if (<= z 1.2e+26) (* (/ z (sqrt (- (* z z) (* t a)))) (* y x)) (* 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 <= -620000.0) {
		tmp = y * -x;
	} else if (z <= 1.2e+26) {
		tmp = (z / sqrt(((z * z) - (t * a)))) * (y * x);
	} 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 <= (-620000.0d0)) then
        tmp = y * -x
    else if (z <= 1.2d+26) then
        tmp = (z / sqrt(((z * z) - (t * a)))) * (y * x)
    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 <= -620000.0) {
		tmp = y * -x;
	} else if (z <= 1.2e+26) {
		tmp = (z / Math.sqrt(((z * z) - (t * a)))) * (y * x);
	} 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 <= -620000.0:
		tmp = y * -x
	elif z <= 1.2e+26:
		tmp = (z / math.sqrt(((z * z) - (t * a)))) * (y * x)
	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 <= -620000.0)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 1.2e+26)
		tmp = Float64(Float64(z / sqrt(Float64(Float64(z * z) - Float64(t * a)))) * Float64(y * x));
	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 <= -620000.0)
		tmp = y * -x;
	elseif (z <= 1.2e+26)
		tmp = (z / sqrt(((z * z) - (t * a)))) * (y * x);
	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, -620000.0], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 1.2e+26], N[(N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Target

Original	24.5
Target	7.9
Herbie	7.7

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

Derivation

1. Split input into 3 regimes

2. if z < -6.2e5

   1. Initial program 33.3

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

   2. Simplified 30.7

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

      [Start] 33.3	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 30.7	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]
      *-commutative [<=] 30.7	\[ \color{blue}{\frac{z}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(x \cdot y\right)} \]

   3. Taylor expanded in z around -inf 5.7

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

   4. Simplified 5.7

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

      [Start] 5.7	\[ -1 \cdot \left(y \cdot x\right) \]
      *-commutative [<=] 5.7	\[ -1 \cdot \color{blue}{\left(x \cdot y\right)} \]
      mul-1-neg [=>] 5.7	\[ \color{blue}{-x \cdot y} \]
      *-commutative [=>] 5.7	\[ -\color{blue}{y \cdot x} \]
      distribute-lft-neg-in [=>] 5.7	\[ \color{blue}{\left(-y\right) \cdot x} \]

   if -6.2e5 < z < 1.20000000000000002e26

   1. Initial program 11.9

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

   2. Simplified 11.1

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

      [Start] 11.9	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 11.1	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]
      *-commutative [<=] 11.1	\[ \color{blue}{\frac{z}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(x \cdot y\right)} \]

   if 1.20000000000000002e26 < z

   1. Initial program 34.8

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

   2. Simplified 31.9

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

      [Start] 34.8	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 31.9	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]
      *-commutative [<=] 31.9	\[ \color{blue}{\frac{z}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(x \cdot y\right)} \]

   3. Taylor expanded in z around inf 4.5

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

4. Final simplification 7.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -620000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Error	7.6
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -620000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+26}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	7.8
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -620000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Error	12.4
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-133}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-69}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{z + -0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\\ \end{array} \]

Alternative 4
Error	12.5
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-129}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-69}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{t \cdot \left(-a\right)}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{z + -0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\\ \end{array} \]

Alternative 5
Error	12.4
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-128}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-68}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{z + -0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\\ \end{array} \]

Alternative 6
Error	16.9
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{-132}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-93}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{z + -0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7
Error	17.3
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-136}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-159}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8
Error	17.3
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-130}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-161}:\\ \;\;\;\;-2 \cdot \left(\frac{z \cdot \left(z \cdot x\right)}{t} \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 9
Error	17.3
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-136}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-167}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(\frac{-2}{t} \cdot \left(z \cdot \frac{y}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10
Error	17.2
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-133}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-166}:\\ \;\;\;\;y \cdot \left(\frac{-2}{a} \cdot \frac{z \cdot \left(z \cdot x\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11
Error	16.9
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-132}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + -0.5 \cdot \frac{t \cdot a}{z}}{z}}\\ \end{array} \]

Alternative 12
Error	16.1
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-135}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{z + -0.5 \cdot \left(t \cdot \frac{a}{z}\right)}\\ \end{array} \]

Alternative 13
Error	18.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-269}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{\frac{z}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 14
Error	17.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-133}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-171}:\\ \;\;\;\;\left(1 - y \cdot x\right) + -1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 15
Error	19.7
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-273}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 16
Error	36.7
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023018 
(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)))))