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

Percentage Accurate: 61.8% → 89.7%

Time: 24.5s

Precision: binary64

Cost: 7496

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

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{a}{\frac{z \cdot z}{t}}\\ \mathbf{if}\;z \leq -7.3 \cdot 10^{+87}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, t_1, -1\right)}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + t_1 \cdot -0.5}\\ \end{array} \]

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
 (let* ((t_1 (/ a (/ (* z z) t))))
   (if (<= z -7.3e+87)
     (/ (* x y) (fma 0.5 t_1 -1.0))
     (if (<= z 1.32e+31)
       (/ x (/ (sqrt (- (* z z) (* a t))) (* z y)))
       (/ (* x y) (+ 1.0 (* t_1 -0.5)))))))

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 t_1 = a / ((z * z) / t);
	double tmp;
	if (z <= -7.3e+87) {
		tmp = (x * y) / fma(0.5, t_1, -1.0);
	} else if (z <= 1.32e+31) {
		tmp = x / (sqrt(((z * z) - (a * t))) / (z * y));
	} else {
		tmp = (x * y) / (1.0 + (t_1 * -0.5));
	}
	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)
	t_1 = Float64(a / Float64(Float64(z * z) / t))
	tmp = 0.0
	if (z <= -7.3e+87)
		tmp = Float64(Float64(x * y) / fma(0.5, t_1, -1.0));
	elseif (z <= 1.32e+31)
		tmp = Float64(x / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / Float64(z * y)));
	else
		tmp = Float64(Float64(x * y) / Float64(1.0 + Float64(t_1 * -0.5)));
	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_] := Block[{t$95$1 = N[(a / N[(N[(z * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.3e+87], N[(N[(x * y), $MachinePrecision] / N[(0.5 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.32e+31], N[(x / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[(1.0 + N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	61.8%
Target	88.9%
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 < -7.29999999999999997e87

   1. Initial program 22.7%

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

   2. Simplified 25.1%

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

      [Start] 22.7%	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-/l* [=>] 25.1%	\[ \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Taylor expanded in z around -inf 85.3%

      \[\leadsto \frac{x \cdot y}{\color{blue}{0.5 \cdot \frac{a \cdot t}{{z}^{2}} - 1}} \]

   4. Simplified 95.9%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z \cdot z}{t}}, -1\right)}} \]
      Step-by-step derivation

      [Start] 85.3%	\[ \frac{x \cdot y}{0.5 \cdot \frac{a \cdot t}{{z}^{2}} - 1} \]
      fma-neg [=>] 85.3%	\[ \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a \cdot t}{{z}^{2}}, -1\right)}} \]
      unpow2 [=>] 85.3%	\[ \frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a \cdot t}{\color{blue}{z \cdot z}}, -1\right)} \]
      associate-/l* [=>] 95.9%	\[ \frac{x \cdot y}{\mathsf{fma}\left(0.5, \color{blue}{\frac{a}{\frac{z \cdot z}{t}}}, -1\right)} \]
      metadata-eval [=>] 95.9%	\[ \frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z \cdot z}{t}}, \color{blue}{-1}\right)} \]

   if -7.29999999999999997e87 < z < 1.32000000000000011e31

   1. Initial program 82.8%

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

   2. Simplified 82.9%

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

      [Start] 82.8%	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      *-commutative [=>] 82.8%	\[ \frac{\color{blue}{\left(y \cdot x\right)} \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*l* [=>] 83.1%	\[ \frac{\color{blue}{y \cdot \left(x \cdot z\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 82.9%	\[ \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Applied egg-rr 82.2%

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

      [Start] 82.9%	\[ y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      *-commutative [=>] 82.9%	\[ \color{blue}{\frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}} \cdot y} \]
      associate-/l* [=>] 82.2%	\[ \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \cdot y \]
      associate-/r/ [<=] 82.2%	\[ \color{blue}{\frac{x}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]
      associate-/l/ [=>] 82.2%	\[ \frac{x}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{y \cdot z}}} \]
      *-commutative [=>] 82.2%	\[ \frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{\color{blue}{z \cdot y}}} \]

   if 1.32000000000000011e31 < z

   1. Initial program 30.0%

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

   2. Simplified 32.8%

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

      [Start] 30.0%	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-/l* [=>] 32.8%	\[ \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Taylor expanded in z around inf 93.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}}} \]

   4. Simplified 97.7%

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 + -0.5 \cdot \frac{a}{\frac{z \cdot z}{t}}}} \]
      Step-by-step derivation

      [Start] 93.8%	\[ \frac{x \cdot y}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}} \]
      unpow2 [=>] 93.8%	\[ \frac{x \cdot y}{1 + -0.5 \cdot \frac{a \cdot t}{\color{blue}{z \cdot z}}} \]
      associate-/l* [=>] 97.7%	\[ \frac{x \cdot y}{1 + -0.5 \cdot \color{blue}{\frac{a}{\frac{z \cdot z}{t}}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.3 \cdot 10^{+87}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{a}{\frac{z \cdot z}{t}}, -1\right)}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.7%
Cost	7496

\[\begin{array}{l} t_1 := \frac{a}{\frac{z \cdot z}{t}}\\ \mathbf{if}\;z \leq -7.3 \cdot 10^{+87}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, t_1, -1\right)}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + t_1 \cdot -0.5}\\ \end{array} \]

Alternative 2
Accuracy	89.5%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+112}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+112}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3
Accuracy	89.8%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+92}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+110}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4
Accuracy	89.6%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 5
Accuracy	82.5%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-149}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 6
Accuracy	82.8%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-143}:\\ \;\;\;\;\frac{z \cdot x}{\frac{\sqrt{a \cdot \left(-t\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 7
Accuracy	82.8%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 8
Accuracy	77.1%
Cost	1356

\[\begin{array}{l} t_1 := \frac{a \cdot t}{z}\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+92}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-286}:\\ \;\;\;\;z \cdot \frac{x \cdot y}{0.5 \cdot t_1 - z}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+105}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{z + -0.5 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9
Accuracy	76.3%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+131}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-286}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{0.5 \cdot \frac{a \cdot t}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a \cdot \left(0.5 \cdot \frac{t}{z}\right)}{z}}\\ \end{array} \]

Alternative 10
Accuracy	73.3%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-206}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11
Accuracy	73.7%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-277}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-181}:\\ \;\;\;\;\frac{z \cdot y}{0.5} \cdot \frac{x}{a \cdot \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 12
Accuracy	73.8%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-181}:\\ \;\;\;\;\frac{x \cdot y}{\frac{a \cdot \left(0.5 \cdot \frac{t}{z}\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 13
Accuracy	74.7%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a}{\frac{z \cdot z}{t}} \cdot -0.5}\\ \end{array} \]

Alternative 14
Accuracy	74.7%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-75}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{1 + \frac{a \cdot \left(0.5 \cdot \frac{t}{z}\right)}{z}}\\ \end{array} \]

Alternative 15
Accuracy	74.5%
Cost	776

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-213}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot \left(-y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 16
Accuracy	72.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq 2.25 \cdot 10^{-302}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-149}:\\ \;\;\;\;\frac{z \cdot y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 17
Accuracy	73.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-83}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 18
Accuracy	71.7%
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-286}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 19
Accuracy	42.8%
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2023272 
(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)))))