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

Average Accuracy: 61.4% → 89.7%

Time: 20.4s

Precision: binary64

Cost: 7496

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-185}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}}\\ \end{array} \]

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 -3.5e+26)
   (* y (- x))
   (if (<= z 2.4e-185)
     (/ x (/ (sqrt (- (* z z) (* a t))) (* z y)))
     (/ (* y x) (sqrt (- 1.0 (* (/ t z) (/ a z))))))))

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 <= -3.5e+26) {
		tmp = y * -x;
	} else if (z <= 2.4e-185) {
		tmp = x / (sqrt(((z * z) - (a * t))) / (z * y));
	} else {
		tmp = (y * x) / sqrt((1.0 - ((t / z) * (a / z))));
	}
	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 <= (-3.5d+26)) then
        tmp = y * -x
    else if (z <= 2.4d-185) then
        tmp = x / (sqrt(((z * z) - (a * t))) / (z * y))
    else
        tmp = (y * x) / sqrt((1.0d0 - ((t / z) * (a / z))))
    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 <= -3.5e+26) {
		tmp = y * -x;
	} else if (z <= 2.4e-185) {
		tmp = x / (Math.sqrt(((z * z) - (a * t))) / (z * y));
	} else {
		tmp = (y * x) / Math.sqrt((1.0 - ((t / z) * (a / z))));
	}
	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 <= -3.5e+26:
		tmp = y * -x
	elif z <= 2.4e-185:
		tmp = x / (math.sqrt(((z * z) - (a * t))) / (z * y))
	else:
		tmp = (y * x) / math.sqrt((1.0 - ((t / z) * (a / z))))
	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 <= -3.5e+26)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 2.4e-185)
		tmp = Float64(x / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / Float64(z * y)));
	else
		tmp = Float64(Float64(y * x) / sqrt(Float64(1.0 - Float64(Float64(t / z) * Float64(a / z)))));
	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 <= -3.5e+26)
		tmp = y * -x;
	elseif (z <= 2.4e-185)
		tmp = x / (sqrt(((z * z) - (a * t))) / (z * y));
	else
		tmp = (y * x) / sqrt((1.0 - ((t / z) * (a / z))));
	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, -3.5e+26], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 2.4e-185], N[(x / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(t / z), $MachinePrecision] * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	61.4%
Target	89.0%
Herbie	89.7%

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

Derivation

1. Split input into 3 regimes

2. if z < -3.4999999999999999e26

   1. Initial program 41.2%

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

   2. Simplified 44.5%

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

      [Start] 41.2	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      *-commutative [=>] 41.2	\[ \frac{\color{blue}{\left(y \cdot x\right)} \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*l* [=>] 40.9	\[ \frac{\color{blue}{y \cdot \left(x \cdot z\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 44.5	\[ \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Taylor expanded in z around -inf 97.8%

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

   4. Simplified 97.8%

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

      [Start] 97.8	\[ y \cdot \left(-1 \cdot x\right) \]
      neg-mul-1 [<=] 97.8	\[ y \cdot \color{blue}{\left(-x\right)} \]

   if -3.4999999999999999e26 < z < 2.4000000000000001e-185

   1. Initial program 82.2%

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

   2. Simplified 82.8%

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

      [Start] 82.2	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*l/ [<=] 82.8	\[ \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

   3. Applied egg-rr 86.1%

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

      [Start] 82.8	\[ \frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z \]
      associate-*l/ [=>] 82.2	\[ \color{blue}{\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]
      add-sqr-sqrt [=>] 82.0	\[ \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\sqrt{\sqrt{z \cdot z - t \cdot a}} \cdot \sqrt{\sqrt{z \cdot z - t \cdot a}}}} \]
      associate-/r* [=>] 82.1	\[ \color{blue}{\frac{\frac{\left(x \cdot y\right) \cdot z}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}} \]
      associate-*l* [=>] 86.0	\[ \frac{\frac{\color{blue}{x \cdot \left(y \cdot z\right)}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \]
      pow1/2 [=>] 86.0	\[ \frac{\frac{x \cdot \left(y \cdot z\right)}{\sqrt{\color{blue}{{\left(z \cdot z - t \cdot a\right)}^{0.5}}}}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \]
      sqrt-pow1 [=>] 86.1	\[ \frac{\frac{x \cdot \left(y \cdot z\right)}{\color{blue}{{\left(z \cdot z - t \cdot a\right)}^{\left(\frac{0.5}{2}\right)}}}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \]
      metadata-eval [=>] 86.1	\[ \frac{\frac{x \cdot \left(y \cdot z\right)}{{\left(z \cdot z - t \cdot a\right)}^{\color{blue}{0.25}}}}{\sqrt{\sqrt{z \cdot z - t \cdot a}}} \]
      pow1/2 [=>] 86.1	\[ \frac{\frac{x \cdot \left(y \cdot z\right)}{{\left(z \cdot z - t \cdot a\right)}^{0.25}}}{\sqrt{\color{blue}{{\left(z \cdot z - t \cdot a\right)}^{0.5}}}} \]
      sqrt-pow1 [=>] 86.1	\[ \frac{\frac{x \cdot \left(y \cdot z\right)}{{\left(z \cdot z - t \cdot a\right)}^{0.25}}}{\color{blue}{{\left(z \cdot z - t \cdot a\right)}^{\left(\frac{0.5}{2}\right)}}} \]
      metadata-eval [=>] 86.1	\[ \frac{\frac{x \cdot \left(y \cdot z\right)}{{\left(z \cdot z - t \cdot a\right)}^{0.25}}}{{\left(z \cdot z - t \cdot a\right)}^{\color{blue}{0.25}}} \]

   4. Simplified 84.7%

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

      [Start] 86.1	\[ \frac{\frac{x \cdot \left(y \cdot z\right)}{{\left(z \cdot z - t \cdot a\right)}^{0.25}}}{{\left(z \cdot z - t \cdot a\right)}^{0.25}} \]
      associate-/l/ [=>] 86.0	\[ \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{{\left(z \cdot z - t \cdot a\right)}^{0.25} \cdot {\left(z \cdot z - t \cdot a\right)}^{0.25}}} \]
      associate-/l* [=>] 84.7	\[ \color{blue}{\frac{x}{\frac{{\left(z \cdot z - t \cdot a\right)}^{0.25} \cdot {\left(z \cdot z - t \cdot a\right)}^{0.25}}{y \cdot z}}} \]
      pow-sqr [=>] 84.7	\[ \frac{x}{\frac{\color{blue}{{\left(z \cdot z - t \cdot a\right)}^{\left(2 \cdot 0.25\right)}}}{y \cdot z}} \]
      metadata-eval [=>] 84.7	\[ \frac{x}{\frac{{\left(z \cdot z - t \cdot a\right)}^{\color{blue}{0.5}}}{y \cdot z}} \]
      unpow1/2 [=>] 84.7	\[ \frac{x}{\frac{\color{blue}{\sqrt{z \cdot z - t \cdot a}}}{y \cdot z}} \]
      *-commutative [<=] 84.7	\[ \frac{x}{\frac{\sqrt{z \cdot z - \color{blue}{a \cdot t}}}{y \cdot z}} \]
      *-commutative [=>] 84.7	\[ \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{\color{blue}{z \cdot y}}} \]

   if 2.4000000000000001e-185 < z

   1. Initial program 56.2%

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

   2. Simplified 58.3%

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

      [Start] 56.2	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-/l* [=>] 58.3	\[ \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Applied egg-rr 45.1%

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

      [Start] 58.3	\[ \frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]
      add-sqr-sqrt [=>] 58.3	\[ \frac{x \cdot y}{\color{blue}{\sqrt{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot \sqrt{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}}} \]
      sqrt-unprod [=>] 55.4	\[ \frac{x \cdot y}{\color{blue}{\sqrt{\frac{\sqrt{z \cdot z - t \cdot a}}{z} \cdot \frac{\sqrt{z \cdot z - t \cdot a}}{z}}}} \]
      frac-times [=>] 45.1	\[ \frac{x \cdot y}{\sqrt{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a} \cdot \sqrt{z \cdot z - t \cdot a}}{z \cdot z}}}} \]
      add-sqr-sqrt [<=] 45.1	\[ \frac{x \cdot y}{\sqrt{\frac{\color{blue}{z \cdot z - t \cdot a}}{z \cdot z}}} \]

   4. Simplified 96.1%

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

      [Start] 45.1	\[ \frac{x \cdot y}{\sqrt{\frac{z \cdot z - t \cdot a}{z \cdot z}}} \]
      unpow2 [<=] 45.1	\[ \frac{x \cdot y}{\sqrt{\frac{\color{blue}{{z}^{2}} - t \cdot a}{z \cdot z}}} \]
      unpow2 [<=] 45.1	\[ \frac{x \cdot y}{\sqrt{\frac{{z}^{2} - t \cdot a}{\color{blue}{{z}^{2}}}}} \]
      div-sub [=>] 45.1	\[ \frac{x \cdot y}{\sqrt{\color{blue}{\frac{{z}^{2}}{{z}^{2}} - \frac{t \cdot a}{{z}^{2}}}}} \]
      *-inverses [=>] 84.9	\[ \frac{x \cdot y}{\sqrt{\color{blue}{1} - \frac{t \cdot a}{{z}^{2}}}} \]
      unpow2 [=>] 84.9	\[ \frac{x \cdot y}{\sqrt{1 - \frac{t \cdot a}{\color{blue}{z \cdot z}}}} \]
      times-frac [=>] 96.1	\[ \frac{x \cdot y}{\sqrt{1 - \color{blue}{\frac{t}{z} \cdot \frac{a}{z}}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-185}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.8%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+71}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+59}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Accuracy	89.7%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+53}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+58}:\\ \;\;\;\;z \cdot \frac{y \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Accuracy	89.3%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+27}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+58}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4
Accuracy	81.5%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-102}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\frac{a \cdot 0.5}{\frac{z}{t}} - z}{z}}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-185}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5
Accuracy	81.3%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\frac{a \cdot 0.5}{\frac{z}{t}} - z}{z}}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-185}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{a \cdot \left(-t\right)}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Accuracy	75.4%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+53}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-233}:\\ \;\;\;\;y \cdot \frac{z}{\frac{\frac{t}{\frac{z}{a \cdot 0.5}} - z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7
Accuracy	76.2%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{-233}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\frac{a \cdot 0.5}{\frac{z}{t}} - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8
Accuracy	72.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-205}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-269}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 9
Accuracy	74.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-183}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-56}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10
Accuracy	74.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-117}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-252}:\\ \;\;\;\;\left(1 - y \cdot x\right) + -1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11
Accuracy	72.1%
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-303}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 12
Accuracy	42.0%
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023158 
(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)))))