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

Average Error: 25.0 → 6.1

Time: 36.2s

Precision: binary64

Cost: 7496

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1.202 \cdot 10^{+123}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z}{t}}, -z\right)}\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - t \cdot \frac{a}{z \cdot z}}}\\ \end{array} \]

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 -1.202e+123)
   (* x (* y (/ z (fma 0.5 (/ a (/ z t)) (- z)))))
   (if (<= z 1.1e-29)
     (* x (* y (/ z (sqrt (- (* z z) (* a t))))))
     (/ (* x y) (sqrt (- 1.0 (* t (/ a (* z 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 <= -1.202e+123) {
		tmp = x * (y * (z / fma(0.5, (a / (z / t)), -z)));
	} else if (z <= 1.1e-29) {
		tmp = x * (y * (z / sqrt(((z * z) - (a * t)))));
	} else {
		tmp = (x * y) / sqrt((1.0 - (t * (a / (z * 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 <= -1.202e+123)
		tmp = Float64(x * Float64(y * Float64(z / fma(0.5, Float64(a / Float64(z / t)), Float64(-z)))));
	elseif (z <= 1.1e-29)
		tmp = Float64(x * Float64(y * Float64(z / sqrt(Float64(Float64(z * z) - Float64(a * t))))));
	else
		tmp = Float64(Float64(x * y) / sqrt(Float64(1.0 - Float64(t * Float64(a / Float64(z * z))))));
	end
	return 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, -1.202e+123], N[(x * N[(y * N[(z / N[(0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision] + (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e-29], N[(x * N[(y * N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t * N[(a / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	25.0
Target	7.9
Herbie	6.1

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

Derivation

1. Split input into 3 regimes

2. if z < -1.20199999999999993e123

   1. Initial program 48.0

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

   2. Simplified 45.9

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

      [Start] 48.0	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 45.9	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]
      associate-*l* [=>] 45.9	\[ \color{blue}{x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)} \]

   3. Taylor expanded in z around -inf 5.3

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}}\right) \]

   4. Simplified 1.8

      \[\leadsto x \cdot \left(y \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z}{t}}, -z\right)}}\right) \]
      Proof

      [Start] 5.3	\[ x \cdot \left(y \cdot \frac{z}{0.5 \cdot \frac{a \cdot t}{z} + -1 \cdot z}\right) \]
      mul-1-neg [=>] 5.3	\[ x \cdot \left(y \cdot \frac{z}{0.5 \cdot \frac{a \cdot t}{z} + \color{blue}{\left(-z\right)}}\right) \]
      fma-def [=>] 5.3	\[ x \cdot \left(y \cdot \frac{z}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a \cdot t}{z}, -z\right)}}\right) \]
      associate-/l* [=>] 1.8	\[ x \cdot \left(y \cdot \frac{z}{\mathsf{fma}\left(0.5, \color{blue}{\frac{a}{\frac{z}{t}}}, -z\right)}\right) \]

   if -1.20199999999999993e123 < z < 1.09999999999999995e-29

   1. Initial program 12.5

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

   2. Simplified 10.5

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

      [Start] 12.5	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 11.0	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]
      associate-*l* [=>] 10.5	\[ \color{blue}{x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)} \]

   if 1.09999999999999995e-29 < z

   1. Initial program 31.3

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

   2. Simplified 28.7

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

      [Start] 31.3	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-/l* [=>] 28.7	\[ \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Applied egg-rr 34.2

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

   4. Simplified 1.7

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

      [Start] 34.2	\[ \frac{x \cdot y}{\sqrt{\frac{z \cdot z - t \cdot a}{z \cdot z}}} \]
      *-commutative [=>] 34.2	\[ \frac{x \cdot y}{\sqrt{\frac{z \cdot z - \color{blue}{a \cdot t}}{z \cdot z}}} \]
      div-sub [=>] 34.2	\[ \frac{x \cdot y}{\sqrt{\color{blue}{\frac{z \cdot z}{z \cdot z} - \frac{a \cdot t}{z \cdot z}}}} \]
      unpow2 [<=] 34.2	\[ \frac{x \cdot y}{\sqrt{\frac{\color{blue}{{z}^{2}}}{z \cdot z} - \frac{a \cdot t}{z \cdot z}}} \]
      unpow2 [<=] 34.2	\[ \frac{x \cdot y}{\sqrt{\frac{{z}^{2}}{\color{blue}{{z}^{2}}} - \frac{a \cdot t}{z \cdot z}}} \]
      *-inverses [=>] 5.0	\[ \frac{x \cdot y}{\sqrt{\color{blue}{1} - \frac{a \cdot t}{z \cdot z}}} \]
      unpow2 [<=] 5.0	\[ \frac{x \cdot y}{\sqrt{1 - \frac{a \cdot t}{\color{blue}{{z}^{2}}}}} \]
      associate-/l* [=>] 1.6	\[ \frac{x \cdot y}{\sqrt{1 - \color{blue}{\frac{a}{\frac{{z}^{2}}{t}}}}} \]
      associate-/r/ [=>] 1.7	\[ \frac{x \cdot y}{\sqrt{1 - \color{blue}{\frac{a}{{z}^{2}} \cdot t}}} \]
      unpow2 [=>] 1.7	\[ \frac{x \cdot y}{\sqrt{1 - \frac{a}{\color{blue}{z \cdot z}} \cdot t}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.202 \cdot 10^{+123}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z}{t}}, -z\right)}\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - t \cdot \frac{a}{z \cdot z}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.4
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -1.202 \cdot 10^{+123}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)}\right)\\ \end{array} \]

Alternative 2
Error	6.1
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -1.202 \cdot 10^{+123}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+53}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{1 - t \cdot \frac{a}{z \cdot z}}}{x}}\\ \end{array} \]

Alternative 3
Error	6.1
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -1.202 \cdot 10^{+123}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-30}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - t \cdot \frac{a}{z \cdot z}}}\\ \end{array} \]

Alternative 4
Error	6.1
Cost	7496

\[\begin{array}{l} t_1 := t \cdot \frac{a}{z \cdot z}\\ \mathbf{if}\;z \leq -1.202 \cdot 10^{+123}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, t_1, -1\right)}\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-30}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - t_1}}\\ \end{array} \]

Alternative 5
Error	12.0
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-123}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)}\right)\\ \end{array} \]

Alternative 6
Error	15.2
Cost	1420

\[\begin{array}{l} t_1 := x \cdot \left(-y\right)\\ t_2 := z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)\\ \mathbf{if}\;z \leq -50:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-148}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{-y}{z - a \cdot \left(0.5 \cdot \frac{t}{z}\right)}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{z \cdot t_1}{t_2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{t_2}\right)\\ \end{array} \]

Alternative 7
Error	15.2
Cost	1420

\[\begin{array}{l} \mathbf{if}\;z \leq -950:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{-148}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{-y}{z - a \cdot \left(0.5 \cdot \frac{t}{z}\right)}\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot \left(-y\right)\right)}{z + \left(a \cdot t\right) \cdot \frac{-0.5}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)}\right)\\ \end{array} \]

Alternative 8
Error	15.2
Cost	1420

\[\begin{array}{l} \mathbf{if}\;z \leq -920:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{z \cdot x}{\frac{z + \frac{-0.5 \cdot \left(a \cdot t\right)}{z}}{-y}}\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot \left(-y\right)\right)}{z + \left(a \cdot t\right) \cdot \frac{-0.5}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)}\right)\\ \end{array} \]

Alternative 9
Error	15.2
Cost	1356

\[\begin{array}{l} t_1 := x \cdot \left(-y\right)\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.8 \cdot 10^{-151}:\\ \;\;\;\;\frac{y}{\frac{a \cdot \left(0.5 \cdot \frac{t}{z}\right) - z}{z \cdot x}}\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)}\right)\\ \end{array} \]

Alternative 10
Error	15.3
Cost	1356

\[\begin{array}{l} t_1 := x \cdot \left(-y\right)\\ \mathbf{if}\;z \leq -2.7:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.8 \cdot 10^{-151}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{-y}{z - a \cdot \left(0.5 \cdot \frac{t}{z}\right)}\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)}\right)\\ \end{array} \]

Alternative 11
Error	14.5
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -1.202 \cdot 10^{+123}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-224}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{0.5 \cdot \frac{a \cdot t}{z} - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)}\right)\\ \end{array} \]

Alternative 12
Error	17.0
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-153}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-123}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 13
Error	17.0
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-151}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-133}:\\ \;\;\;\;2 \cdot \left(\frac{y}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 14
Error	17.3
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-151}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-133}:\\ \;\;\;\;2 \cdot \frac{y}{\frac{a \cdot t}{x \cdot \left(z \cdot z\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 15
Error	16.9
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-153}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-124}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-2 \cdot \frac{z}{t \cdot \frac{a}{z}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 16
Error	15.7
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-155}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \left(a \cdot \frac{t}{z}\right)}\right)\\ \end{array} \]

Alternative 17
Error	17.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -3.05 \cdot 10^{-158}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-132}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot \left(-z\right)\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 18
Error	17.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-156}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{-226}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 19
Error	17.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.42 \cdot 10^{-154}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-157}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 20
Error	19.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 21
Error	36.7
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2022364 
(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)))))