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

Average Accuracy: 61.4% → 88.0%

Time: 57.0s

Precision: binary64

Cost: 7560

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t}{z} \cdot \frac{a}{z}\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, t_1, -1\right)}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-220}:\\ \;\;\;\;{\left(z \cdot z - t \cdot a\right)}^{-0.5} \cdot \left(x \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{1 - t_1}}\\ \end{array} \]

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 (* (/ t z) (/ a z))))
   (if (<= z -7.8e+118)
     (/ (* x y) (fma 0.5 t_1 -1.0))
     (if (<= z 3.8e-220)
       (* (pow (- (* z z) (* t a)) -0.5) (* x (* z y)))
       (* y (/ x (sqrt (- 1.0 t_1))))))))

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 = (t / z) * (a / z);
	double tmp;
	if (z <= -7.8e+118) {
		tmp = (x * y) / fma(0.5, t_1, -1.0);
	} else if (z <= 3.8e-220) {
		tmp = pow(((z * z) - (t * a)), -0.5) * (x * (z * y));
	} else {
		tmp = y * (x / sqrt((1.0 - t_1)));
	}
	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(Float64(t / z) * Float64(a / z))
	tmp = 0.0
	if (z <= -7.8e+118)
		tmp = Float64(Float64(x * y) / fma(0.5, t_1, -1.0));
	elseif (z <= 3.8e-220)
		tmp = Float64((Float64(Float64(z * z) - Float64(t * a)) ^ -0.5) * Float64(x * Float64(z * y)));
	else
		tmp = Float64(y * Float64(x / sqrt(Float64(1.0 - t_1))));
	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[(N[(t / z), $MachinePrecision] * N[(a / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.8e+118], N[(N[(x * y), $MachinePrecision] / N[(0.5 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e-220], N[(N[Power[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	61.4%
Target	88.3%
Herbie	88.0%

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

Derivation

1. Split input into 3 regimes

2. if z < -7.8e118

   1. Initial program 27.6%

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

   2. Simplified 30.7%

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

      [Start] 27.6	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-/l* [=>] 30.7	\[ \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Taylor expanded in z around -inf 90.0%

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

   4. Simplified 97.6%

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

      [Start] 90.0	\[ \frac{x \cdot y}{0.5 \cdot \frac{a \cdot t}{{z}^{2}} - 1} \]
      fma-neg [=>] 90.0	\[ \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a \cdot t}{{z}^{2}}, -1\right)}} \]
      *-commutative [=>] 90.0	\[ \frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{\color{blue}{t \cdot a}}{{z}^{2}}, -1\right)} \]
      unpow2 [=>] 90.0	\[ \frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{t \cdot a}{\color{blue}{z \cdot z}}, -1\right)} \]
      times-frac [=>] 97.6	\[ \frac{x \cdot y}{\mathsf{fma}\left(0.5, \color{blue}{\frac{t}{z} \cdot \frac{a}{z}}, -1\right)} \]
      metadata-eval [=>] 97.6	\[ \frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{t}{z} \cdot \frac{a}{z}, \color{blue}{-1}\right)} \]

   if -7.8e118 < z < 3.80000000000000009e-220

   1. Initial program 81.7%

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

   2. Applied egg-rr 78.9%

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

      [Start] 81.7	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      clear-num [=>] 81.2	\[ \color{blue}{\frac{1}{\frac{\sqrt{z \cdot z - t \cdot a}}{\left(x \cdot y\right) \cdot z}}} \]
      associate-/r/ [=>] 81.7	\[ \color{blue}{\frac{1}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(\left(x \cdot y\right) \cdot z\right)} \]
      pow1/2 [=>] 81.7	\[ \frac{1}{\color{blue}{{\left(z \cdot z - t \cdot a\right)}^{0.5}}} \cdot \left(\left(x \cdot y\right) \cdot z\right) \]
      pow-flip [=>] 81.7	\[ \color{blue}{{\left(z \cdot z - t \cdot a\right)}^{\left(-0.5\right)}} \cdot \left(\left(x \cdot y\right) \cdot z\right) \]
      metadata-eval [=>] 81.7	\[ {\left(z \cdot z - t \cdot a\right)}^{\color{blue}{-0.5}} \cdot \left(\left(x \cdot y\right) \cdot z\right) \]
      associate-*l* [=>] 78.9	\[ {\left(z \cdot z - t \cdot a\right)}^{-0.5} \cdot \color{blue}{\left(x \cdot \left(y \cdot z\right)\right)} \]

   if 3.80000000000000009e-220 < z

   1. Initial program 59.9%

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

   2. Simplified 63.7%

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

      [Start] 59.9	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-/l* [=>] 63.7	\[ \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Applied egg-rr 53.3%

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

      [Start] 63.7	\[ \frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]
      add-sqr-sqrt [=>] 63.7	\[ \frac{x \cdot y}{\color{blue}{\sqrt{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot \sqrt{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}}} \]
      sqrt-unprod [=>] 59.8	\[ \frac{x \cdot y}{\color{blue}{\sqrt{\frac{\sqrt{z \cdot z - t \cdot a}}{z} \cdot \frac{\sqrt{z \cdot z - t \cdot a}}{z}}}} \]
      frac-times [=>] 53.2	\[ \frac{x \cdot y}{\sqrt{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a} \cdot \sqrt{z \cdot z - t \cdot a}}{z \cdot z}}}} \]
      add-sqr-sqrt [<=] 53.3	\[ \frac{x \cdot y}{\sqrt{\frac{\color{blue}{z \cdot z - t \cdot a}}{z \cdot z}}} \]

   4. Simplified 90.7%

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

      [Start] 53.3	\[ \frac{x \cdot y}{\sqrt{\frac{z \cdot z - t \cdot a}{z \cdot z}}} \]
      unpow2 [<=] 53.3	\[ \frac{x \cdot y}{\sqrt{\frac{\color{blue}{{z}^{2}} - t \cdot a}{z \cdot z}}} \]
      unpow2 [<=] 53.3	\[ \frac{x \cdot y}{\sqrt{\frac{{z}^{2} - t \cdot a}{\color{blue}{{z}^{2}}}}} \]
      div-sub [=>] 51.0	\[ \frac{x \cdot y}{\sqrt{\color{blue}{\frac{{z}^{2}}{{z}^{2}} - \frac{t \cdot a}{{z}^{2}}}}} \]
      *-inverses [=>] 84.5	\[ \frac{x \cdot y}{\sqrt{\color{blue}{1} - \frac{t \cdot a}{{z}^{2}}}} \]
      unpow2 [=>] 84.5	\[ \frac{x \cdot y}{\sqrt{1 - \frac{t \cdot a}{\color{blue}{z \cdot z}}}} \]
      times-frac [=>] 90.7	\[ \frac{x \cdot y}{\sqrt{1 - \color{blue}{\frac{t}{z} \cdot \frac{a}{z}}}} \]

   5. Applied egg-rr 91.1%

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

      [Start] 90.7	\[ \frac{x \cdot y}{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}} \]
      associate-/l* [=>] 90.6	\[ \color{blue}{\frac{x}{\frac{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}}{y}}} \]
      associate-/r/ [=>] 91.1	\[ \color{blue}{\frac{x}{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}} \cdot y} \]

3. Recombined 3 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, \frac{t}{z} \cdot \frac{a}{z}, -1\right)}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-220}:\\ \;\;\;\;{\left(z \cdot z - t \cdot a\right)}^{-0.5} \cdot \left(x \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.6%
Cost	7560

\[\begin{array}{l} t_1 := \frac{t}{z} \cdot \frac{a}{z}\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{+126}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, t_1, -1\right)}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-242}:\\ \;\;\;\;y \cdot \left(x \cdot \left(z \cdot {\left(z \cdot z - t \cdot a\right)}^{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{1 - t_1}}\\ \end{array} \]

Alternative 2
Accuracy	84.5%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\frac{0.5 \cdot a}{\frac{z}{t}} - z}{z}}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-153}:\\ \;\;\;\;y \cdot \left(x \cdot \left(z \cdot {\left(t \cdot \left(-a\right)\right)}^{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{\sqrt{1 - a \cdot \frac{t}{z \cdot z}}}\\ \end{array} \]

Alternative 3
Accuracy	89.3%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+68}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\frac{0.5 \cdot a}{\frac{z}{t}} - z}{z}}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-145}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{\sqrt{1 - a \cdot \frac{t}{z \cdot z}}}\\ \end{array} \]

Alternative 4
Accuracy	89.4%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+70}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\frac{0.5 \cdot a}{\frac{z}{t}} - z}{z}}\\ \mathbf{elif}\;z \leq 1.66 \cdot 10^{-224}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}}\\ \end{array} \]

Alternative 5
Accuracy	89.0%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+159}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.7 \cdot 10^{-254}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}}\\ \end{array} \]

Alternative 6
Accuracy	89.5%
Cost	7496

\[\begin{array}{l} t_1 := \frac{t}{z} \cdot \frac{a}{z}\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+127}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(0.5, t_1, -1\right)}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-255}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\sqrt{1 - t_1}}\\ \end{array} \]

Alternative 7
Accuracy	81.3%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\frac{0.5 \cdot a}{\frac{z}{t}} - z}{z}}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-77}:\\ \;\;\;\;y \cdot \left(x \cdot \left(z \cdot {\left(t \cdot \left(-a\right)\right)}^{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 8
Accuracy	81.4%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\frac{0.5 \cdot a}{\frac{z}{t}} - z}{z}}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-76}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 9
Accuracy	74.8%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{x}{t \cdot \left(0.5 \cdot \frac{a}{z}\right) - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 10
Accuracy	74.5%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-153}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+60}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11
Accuracy	75.0%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\frac{t}{z} \cdot \left(0.5 \cdot a\right) - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 12
Accuracy	75.0%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\frac{0.5 \cdot a}{\frac{z}{t}} - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 13
Accuracy	73.5%
Cost	1160

\[\begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{-153}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-181}:\\ \;\;\;\;y \cdot \frac{\frac{2}{a} \cdot \left(z \cdot \left(z \cdot x\right)\right)}{-t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 14
Accuracy	73.4%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-150}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-179}:\\ \;\;\;\;-2 \cdot \left(\frac{x}{\frac{t}{z \cdot z}} \cdot \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 15
Accuracy	73.4%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-145}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 10^{-178}:\\ \;\;\;\;2 \cdot \left(\frac{x}{\frac{a}{z \cdot z}} \cdot \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 16
Accuracy	73.5%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-159}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-179}:\\ \;\;\;\;y \cdot \left(-2 \cdot \left(\frac{z \cdot z}{a} \cdot \frac{x}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 17
Accuracy	73.5%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(2 \cdot \frac{\frac{x}{\frac{a}{z \cdot z}}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 18
Accuracy	73.5%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-147}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-179}:\\ \;\;\;\;y \cdot \left(\frac{2}{a} \cdot \frac{x}{\frac{t}{z \cdot z}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 19
Accuracy	73.6%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-152}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-179}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot \left(\left(z \cdot x\right) \cdot \frac{-2}{a}\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 20
Accuracy	75.1%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-151}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 21
Accuracy	77.0%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\frac{0.5 \cdot a}{\frac{z}{t}} - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 22
Accuracy	72.9%
Cost	776

\[\begin{array}{l} \mathbf{if}\;z \leq -1.42 \cdot 10^{+19}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.85 \cdot 10^{-197}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 23
Accuracy	71.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-235}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-1 + \left(x + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 24
Accuracy	72.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-235}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-197}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 25
Accuracy	69.9%
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-308}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 26
Accuracy	42.3%
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2023147 
(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)))))