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

Average Error: 24.7 → 5.8

Time: 2.2min

Precision: binary64

Cost: 20292

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(-0.5, \frac{a}{\left|z\right|} \cdot \frac{t}{z}, \frac{\left|z\right|}{z}\right)}\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-70}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{x}{\mathsf{hypot}\left(z, \sqrt{a \cdot \left(-t\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}}\\ \end{array} \]

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 -2e+57)
   (/ (* x y) (fma -0.5 (* (/ a (fabs z)) (/ t z)) (/ (fabs z) z)))
   (if (<= z -9.5e-165)
     (/ (* x y) (/ (sqrt (- (* z z) (* a t))) z))
     (if (<= z 6e-70)
       (* y (* z (/ x (hypot z (sqrt (* a (- t)))))))
       (/ (* x y) (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 <= -2e+57) {
		tmp = (x * y) / fma(-0.5, ((a / fabs(z)) * (t / z)), (fabs(z) / z));
	} else if (z <= -9.5e-165) {
		tmp = (x * y) / (sqrt(((z * z) - (a * t))) / z);
	} else if (z <= 6e-70) {
		tmp = y * (z * (x / hypot(z, sqrt((a * -t)))));
	} else {
		tmp = (x * y) / 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 <= -2e+57)
		tmp = Float64(Float64(x * y) / fma(-0.5, Float64(Float64(a / abs(z)) * Float64(t / z)), Float64(abs(z) / z)));
	elseif (z <= -9.5e-165)
		tmp = Float64(Float64(x * y) / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / z));
	elseif (z <= 6e-70)
		tmp = Float64(y * Float64(z * Float64(x / hypot(z, sqrt(Float64(a * Float64(-t)))))));
	else
		tmp = Float64(Float64(x * y) / sqrt(Float64(1.0 - Float64(Float64(t / z) * Float64(a / 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, -2e+57], N[(N[(x * y), $MachinePrecision] / N[(-0.5 * N[(N[(a / N[Abs[z], $MachinePrecision]), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[z], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -9.5e-165], N[(N[(x * y), $MachinePrecision] / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6e-70], N[(y * N[(z * N[(x / N[Sqrt[z ^ 2 + N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(t / z), $MachinePrecision] * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Target

Original	24.7
Target	7.8
Herbie	5.8

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

Derivation

1. Split input into 4 regimes

2. if z < -2.0000000000000001e57

   1. Initial program 37.4

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

   2. Simplified 34.1

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]
      Proof
      (/.f64 (*.f64 x y) (/.f64 (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))) z)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))): 42 points increase in error, 13 points decrease in error

   3. Taylor expanded in t around 0 35.5

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

   4. Simplified 2.7

      \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{a}{\left|z\right|} \cdot \frac{t}{z}, \frac{\left|z\right|}{z}\right)}} \]
      Proof
      (fma.f64 -1/2 (*.f64 (/.f64 a (fabs.f64 z)) (/.f64 t z)) (/.f64 (fabs.f64 z) z)): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (*.f64 (/.f64 a (Rewrite<= rem-sqrt-square_binary64 (sqrt.f64 (*.f64 z z)))) (/.f64 t z)) (/.f64 (fabs.f64 z) z)): 11 points increase in error, 1 points decrease in error
      (fma.f64 -1/2 (*.f64 (/.f64 a (sqrt.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)))) (/.f64 t z)) (/.f64 (fabs.f64 z) z)): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 a t) (*.f64 (sqrt.f64 (pow.f64 z 2)) z))) (/.f64 (fabs.f64 z) z)): 14 points increase in error, 7 points decrease in error
      (fma.f64 -1/2 (/.f64 (*.f64 a t) (*.f64 (sqrt.f64 (pow.f64 z 2)) z)) (/.f64 (Rewrite<= rem-sqrt-square_binary64 (sqrt.f64 (*.f64 z z))) z)): 80 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 (*.f64 a t) (*.f64 (sqrt.f64 (pow.f64 z 2)) z)) (/.f64 (sqrt.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2))) z)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/2 (/.f64 (*.f64 a t) (*.f64 (sqrt.f64 (pow.f64 z 2)) z))) (/.f64 (sqrt.f64 (pow.f64 z 2)) z))): 0 points increase in error, 0 points decrease in error

   if -2.0000000000000001e57 < z < -9.49999999999999973e-165

   1. Initial program 7.9

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

   2. Simplified 6.0

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]
      Proof
      (/.f64 (*.f64 x y) (/.f64 (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))) z)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))): 42 points increase in error, 13 points decrease in error

   if -9.49999999999999973e-165 < z < 6.0000000000000003e-70

   1. Initial program 16.9

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

   2. Simplified 16.1

      \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{y \cdot z}}} \]
      Proof
      (/.f64 x (/.f64 (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))) (*.f64 y z))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x (*.f64 y z)) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))): 37 points increase in error, 22 points decrease in error
      (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x y) z)) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))): 6 points increase in error, 54 points decrease in error

   3. Applied egg-rr 14.9

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

   if 6.0000000000000003e-70 < z

   1. Initial program 28.4

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

   2. Simplified 25.8

      \[\leadsto \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]
      Proof
      (/.f64 (*.f64 x y) (/.f64 (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))) z)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))): 42 points increase in error, 13 points decrease in error

   3. Applied egg-rr 30.7

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

   4. Simplified 1.9

      \[\leadsto \frac{x \cdot y}{\color{blue}{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}}} \]
      Proof
      (sqrt.f64 (-.f64 1 (*.f64 (/.f64 t z) (/.f64 a z)))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (-.f64 (Rewrite<= *-inverses_binary64 (/.f64 (*.f64 z z) (*.f64 z z))) (*.f64 (/.f64 t z) (/.f64 a z)))): 94 points increase in error, 0 points decrease in error
      (sqrt.f64 (-.f64 (/.f64 (*.f64 z z) (*.f64 z z)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 t a) (*.f64 z z))))): 8 points increase in error, 5 points decrease in error
      (sqrt.f64 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 z z) (*.f64 t a)) (*.f64 z z)))): 0 points increase in error, 1 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+57}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(-0.5, \frac{a}{\left|z\right|} \cdot \frac{t}{z}, \frac{\left|z\right|}{z}\right)}\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-70}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{x}{\mathsf{hypot}\left(z, \sqrt{a \cdot \left(-t\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.0
Cost	13900

\[\begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+57}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -7.1 \cdot 10^{-165}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-77}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{x}{\mathsf{hypot}\left(z, \sqrt{a \cdot \left(-t\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}}\\ \end{array} \]

Alternative 2
Error	7.6
Cost	7760

\[\begin{array}{l} t_1 := x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+57}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-226}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{x}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 3
Error	7.6
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{+51}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+60}:\\ \;\;\;\;\frac{y}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 4
Error	7.6
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-250}:\\ \;\;\;\;\frac{y}{\sqrt{z \cdot z - a \cdot t}} \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - \frac{t}{z} \cdot \frac{a}{z}}}\\ \end{array} \]

Alternative 5
Error	12.4
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-157}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-76}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 6
Error	12.1
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-135}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-80}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 7
Error	12.5
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-157}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-80}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{x}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 8
Error	16.5
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-167}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-90}:\\ \;\;\;\;\frac{x}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9
Error	16.5
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-168}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{\frac{z + -0.5 \cdot \frac{a \cdot t}{z}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 10
Error	17.1
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-168}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-111}:\\ \;\;\;\;-2 \cdot \left(\frac{z \cdot z}{a} \cdot \frac{x \cdot y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11
Error	17.1
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-167}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{x \cdot y}{t \cdot \frac{a}{z}} \cdot \frac{z}{-0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 12
Error	15.9
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-169}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 13
Error	15.5
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-306}:\\ \;\;\;\;\frac{x \cdot y}{\frac{0.5 \cdot \frac{a \cdot t}{z} - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 14
Error	17.1
Cost	1032

\[\begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-171}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 10^{-110}:\\ \;\;\;\;\frac{-1 + \left(1 + y \cdot \left(z \cdot x\right)\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 15
Error	18.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-217}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-225}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 16
Error	19.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-268}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-111}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 17
Error	17.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-229}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-166}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 18
Error	17.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-158}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-109}:\\ \;\;\;\;-1 + \left(1 - x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 19
Error	19.3
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 20
Error	37.5
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2022325 
(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)))))