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

Average Error: 24.6 → 6.3

Time: 1.3min

Precision: binary64

Cost: 7760

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

Math

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

↓

\[\begin{array}{l} t_1 := \left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+152}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-269}:\\ \;\;\;\;\frac{y}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

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 (* (* x y) (/ z (sqrt (- (* z z) (* t a)))))))
   (if (<= z -1e+152)
     (* y (- x))
     (if (<= z -1.8e-166)
       t_1
       (if (<= z 4e-269)
         (* (/ y (sqrt (* t (- a)))) (* x z))
         (if (<= z 5.6e+79) t_1 (* y x)))))))

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 = (x * y) * (z / sqrt(((z * z) - (t * a))));
	double tmp;
	if (z <= -1e+152) {
		tmp = y * -x;
	} else if (z <= -1.8e-166) {
		tmp = t_1;
	} else if (z <= 4e-269) {
		tmp = (y / sqrt((t * -a))) * (x * z);
	} else if (z <= 5.6e+79) {
		tmp = t_1;
	} else {
		tmp = y * x;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (x * y) * (z / sqrt(((z * z) - (t * a))))
    if (z <= (-1d+152)) then
        tmp = y * -x
    else if (z <= (-1.8d-166)) then
        tmp = t_1
    else if (z <= 4d-269) then
        tmp = (y / sqrt((t * -a))) * (x * z)
    else if (z <= 5.6d+79) then
        tmp = t_1
    else
        tmp = y * x
    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 t_1 = (x * y) * (z / Math.sqrt(((z * z) - (t * a))));
	double tmp;
	if (z <= -1e+152) {
		tmp = y * -x;
	} else if (z <= -1.8e-166) {
		tmp = t_1;
	} else if (z <= 4e-269) {
		tmp = (y / Math.sqrt((t * -a))) * (x * z);
	} else if (z <= 5.6e+79) {
		tmp = t_1;
	} else {
		tmp = y * x;
	}
	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):
	t_1 = (x * y) * (z / math.sqrt(((z * z) - (t * a))))
	tmp = 0
	if z <= -1e+152:
		tmp = y * -x
	elif z <= -1.8e-166:
		tmp = t_1
	elif z <= 4e-269:
		tmp = (y / math.sqrt((t * -a))) * (x * z)
	elif z <= 5.6e+79:
		tmp = t_1
	else:
		tmp = y * x
	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(x * y) * Float64(z / sqrt(Float64(Float64(z * z) - Float64(t * a)))))
	tmp = 0.0
	if (z <= -1e+152)
		tmp = Float64(y * Float64(-x));
	elseif (z <= -1.8e-166)
		tmp = t_1;
	elseif (z <= 4e-269)
		tmp = Float64(Float64(y / sqrt(Float64(t * Float64(-a)))) * Float64(x * z));
	elseif (z <= 5.6e+79)
		tmp = t_1;
	else
		tmp = Float64(y * x);
	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)
	t_1 = (x * y) * (z / sqrt(((z * z) - (t * a))));
	tmp = 0.0;
	if (z <= -1e+152)
		tmp = y * -x;
	elseif (z <= -1.8e-166)
		tmp = t_1;
	elseif (z <= 4e-269)
		tmp = (y / sqrt((t * -a))) * (x * z);
	elseif (z <= 5.6e+79)
		tmp = t_1;
	else
		tmp = y * x;
	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_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] * N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+152], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, -1.8e-166], t$95$1, If[LessEqual[z, 4e-269], N[(N[(y / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.6e+79], t$95$1, N[(y * x), $MachinePrecision]]]]]]

Target

Original	24.6
Target	8.2
Herbie	6.3

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

Derivation

1. Split input into 4 regimes

2. if z < -1e152

   1. Initial program 53.9

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

   2. Simplified 53.6

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

      [Start] 53.9	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      rational.json-simplify-49 [=>] 53.5	\[ \color{blue}{z \cdot \frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}}} \]
      rational.json-simplify-2 [=>] 53.5	\[ z \cdot \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \]
      rational.json-simplify-49 [=>] 53.7	\[ z \cdot \color{blue}{\left(x \cdot \frac{y}{\sqrt{z \cdot z - t \cdot a}}\right)} \]
      rational.json-simplify-43 [=>] 53.6	\[ \color{blue}{x \cdot \left(\frac{y}{\sqrt{z \cdot z - t \cdot a}} \cdot z\right)} \]
      rational.json-simplify-2 [=>] 53.6	\[ x \cdot \color{blue}{\left(z \cdot \frac{y}{\sqrt{z \cdot z - t \cdot a}}\right)} \]

   3. Taylor expanded in z around -inf 1.3

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

   4. Simplified 1.3

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

      [Start] 1.3	\[ -1 \cdot \left(y \cdot x\right) \]
      rational.json-simplify-43 [=>] 1.3	\[ \color{blue}{y \cdot \left(x \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 1.3	\[ y \cdot \color{blue}{\left(-x\right)} \]

   if -1e152 < z < -1.8e-166 or 3.9999999999999998e-269 < z < 5.6000000000000002e79

   1. Initial program 9.4

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

   2. Simplified 6.6

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

      [Start] 9.4	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      rational.json-simplify-2 [=>] 9.4	\[ \frac{\color{blue}{z \cdot \left(x \cdot y\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      rational.json-simplify-49 [=>] 6.6	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]

   if -1.8e-166 < z < 3.9999999999999998e-269

   1. Initial program 19.1

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

   2. Taylor expanded in z around 0 64.0

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

   3. Simplified 19.1

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

      [Start] 64.0	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-1} \cdot \sqrt{a \cdot t}} \]
      exponential.json-simplify-20 [=>] 19.1	\[ \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{\sqrt{\left(a \cdot t\right) \cdot -1}}} \]
      rational.json-simplify-2 [<=] 19.1	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right)}}} \]
      rational.json-simplify-2 [=>] 19.1	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{-1 \cdot \color{blue}{\left(t \cdot a\right)}}} \]
      rational.json-simplify-43 [=>] 19.1	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{t \cdot \left(a \cdot -1\right)}}} \]
      rational.json-simplify-9 [=>] 19.1	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{t \cdot \color{blue}{\left(-a\right)}}} \]

   4. Applied egg-rr 19.3

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

   if 5.6000000000000002e79 < z

   1. Initial program 41.0

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

   2. Simplified 40.5

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

      [Start] 41.0	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      rational.json-simplify-49 [=>] 40.6	\[ \color{blue}{z \cdot \frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}}} \]
      rational.json-simplify-2 [=>] 40.6	\[ z \cdot \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \]
      rational.json-simplify-49 [=>] 42.0	\[ z \cdot \color{blue}{\left(x \cdot \frac{y}{\sqrt{z \cdot z - t \cdot a}}\right)} \]
      rational.json-simplify-43 [=>] 40.5	\[ \color{blue}{x \cdot \left(\frac{y}{\sqrt{z \cdot z - t \cdot a}} \cdot z\right)} \]
      rational.json-simplify-2 [=>] 40.5	\[ x \cdot \color{blue}{\left(z \cdot \frac{y}{\sqrt{z \cdot z - t \cdot a}}\right)} \]

   3. Taylor expanded in z around inf 2.8

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+152}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-166}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-269}:\\ \;\;\;\;\frac{y}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+79}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Error	9.8
Cost	7628

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\left(-z\right) + \frac{t}{z} \cdot \frac{a}{2}}{z}}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-208}:\\ \;\;\;\;\left(x \cdot \frac{z}{\sqrt{t \cdot \left(-a\right)}}\right) \cdot y\\ \mathbf{elif}\;z \leq 4000000:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	9.2
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\left(-z\right) + \frac{t}{z} \cdot \frac{a}{2}}{z}}\\ \mathbf{elif}\;z \leq 4000000:\\ \;\;\;\;z \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Error	11.9
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-99}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\left(-z\right) + \frac{t}{z} \cdot \frac{a}{2}}{z}}\\ \mathbf{elif}\;z \leq 7.7 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4
Error	12.0
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-103}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\left(-z\right) + \frac{t}{z} \cdot \frac{a}{2}}{z}}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-46}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5
Error	12.0
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{-103}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\left(-z\right) + \frac{t}{z} \cdot \frac{a}{2}}{z}}\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{-46}:\\ \;\;\;\;\left(x \cdot \frac{z}{\sqrt{t \cdot \left(-a\right)}}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Error	12.1
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\left(-z\right) + \frac{t}{z} \cdot \frac{a}{2}}{z}}\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{y}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7
Error	16.2
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-169}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4000000:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{z + -0.5 \cdot \frac{a \cdot t}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8
Error	16.1
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-169}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3250000:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{z + t \cdot \left(\frac{a}{z} \cdot -0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 9
Error	17.1
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+40}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 7.7 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{a \cdot \frac{\frac{t}{z}}{2} - z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10
Error	16.7
Cost	1156

\[\begin{array}{l} \mathbf{if}\;z \leq 7.7 \cdot 10^{-47}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\frac{t}{\frac{2}{\frac{a}{z}}} + \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11
Error	16.7
Cost	1156

\[\begin{array}{l} \mathbf{if}\;z \leq 7.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{x \cdot y}{\frac{t \cdot \left(\frac{a}{z} \cdot 0.5\right) + \left(-z\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 12
Error	18.0
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-193}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 7.7 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-2 \cdot \left(\frac{\frac{z}{t}}{a} \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 13
Error	18.0
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-193}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 7.7 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-2 \cdot \frac{y \cdot \frac{z}{a}}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 14
Error	18.0
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-193}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 7.7 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \left(z \cdot \left(-2 \cdot \frac{\frac{y}{\frac{a}{z}}}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 15
Error	18.0
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-169}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 7.7 \cdot 10^{-47}:\\ \;\;\;\;z \cdot \left(2 \cdot \left(\left(y \cdot x\right) \cdot \frac{\frac{z}{t}}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 16
Error	17.9
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-169}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 7.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot -2\right)}{t \cdot \frac{a}{y \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 17
Error	16.7
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq 7.7 \cdot 10^{-47}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{a \cdot \frac{\frac{t}{z}}{2} - z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 18
Error	17.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-169}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{\frac{1}{y}}{x \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 19
Error	17.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-169}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 20
Error	19.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-307}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 21
Error	36.5
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023074 
(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)))))