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

Percentage Accurate: 61.7% → 91.2%

Time: 18.6s

Precision: binary64

Cost: 7760

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{if}\;z \leq -3 \cdot 10^{+147}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-207}:\\ \;\;\;\;{\left(t \cdot \left(-a\right)\right)}^{-0.5} \cdot \left(z \cdot \left(y \cdot x\right)\right)\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

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
 (let* ((t_1 (* x (* y (/ z (sqrt (- (* z z) (* t a))))))))
   (if (<= z -3e+147)
     (* y (- x))
     (if (<= z -6e-151)
       t_1
       (if (<= z 2.1e-207)
         (* (pow (* t (- a)) -0.5) (* z (* y x)))
         (if (<= z 6.1e+97)
           t_1
           (/ (* y x) (/ (+ z (* -0.5 (/ a (/ z t)))) 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 t_1 = x * (y * (z / sqrt(((z * z) - (t * a)))));
	double tmp;
	if (z <= -3e+147) {
		tmp = y * -x;
	} else if (z <= -6e-151) {
		tmp = t_1;
	} else if (z <= 2.1e-207) {
		tmp = pow((t * -a), -0.5) * (z * (y * x));
	} else if (z <= 6.1e+97) {
		tmp = t_1;
	} else {
		tmp = (y * x) / ((z + (-0.5 * (a / (z / t)))) / z);
	}
	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 <= (-3d+147)) then
        tmp = y * -x
    else if (z <= (-6d-151)) then
        tmp = t_1
    else if (z <= 2.1d-207) then
        tmp = ((t * -a) ** (-0.5d0)) * (z * (y * x))
    else if (z <= 6.1d+97) then
        tmp = t_1
    else
        tmp = (y * x) / ((z + ((-0.5d0) * (a / (z / t)))) / z)
    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 <= -3e+147) {
		tmp = y * -x;
	} else if (z <= -6e-151) {
		tmp = t_1;
	} else if (z <= 2.1e-207) {
		tmp = Math.pow((t * -a), -0.5) * (z * (y * x));
	} else if (z <= 6.1e+97) {
		tmp = t_1;
	} else {
		tmp = (y * x) / ((z + (-0.5 * (a / (z / t)))) / z);
	}
	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 <= -3e+147:
		tmp = y * -x
	elif z <= -6e-151:
		tmp = t_1
	elif z <= 2.1e-207:
		tmp = math.pow((t * -a), -0.5) * (z * (y * x))
	elif z <= 6.1e+97:
		tmp = t_1
	else:
		tmp = (y * x) / ((z + (-0.5 * (a / (z / t)))) / 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)
	t_1 = Float64(x * Float64(y * Float64(z / sqrt(Float64(Float64(z * z) - Float64(t * a))))))
	tmp = 0.0
	if (z <= -3e+147)
		tmp = Float64(y * Float64(-x));
	elseif (z <= -6e-151)
		tmp = t_1;
	elseif (z <= 2.1e-207)
		tmp = Float64((Float64(t * Float64(-a)) ^ -0.5) * Float64(z * Float64(y * x)));
	elseif (z <= 6.1e+97)
		tmp = t_1;
	else
		tmp = Float64(Float64(y * x) / Float64(Float64(z + Float64(-0.5 * Float64(a / Float64(z / t)))) / z));
	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 <= -3e+147)
		tmp = y * -x;
	elseif (z <= -6e-151)
		tmp = t_1;
	elseif (z <= 2.1e-207)
		tmp = ((t * -a) ^ -0.5) * (z * (y * x));
	elseif (z <= 6.1e+97)
		tmp = t_1;
	else
		tmp = (y * x) / ((z + (-0.5 * (a / (z / t)))) / z);
	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[(x * N[(y * N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3e+147], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, -6e-151], t$95$1, If[LessEqual[z, 2.1e-207], N[(N[Power[N[(t * (-a)), $MachinePrecision], -0.5], $MachinePrecision] * N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.1e+97], t$95$1, N[(N[(y * x), $MachinePrecision] / N[(N[(z + N[(-0.5 * N[(a / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	61.7%
Target	88.9%
Herbie	91.2%

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

Derivation

1. Split input into 4 regimes

2. if z < -2.99999999999999993e147

   1. Initial program 6.9%

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

   2. Simplified 6.6%

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

      [Start] 6.9%	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      *-commutative [=>] 6.9%	\[ \frac{\color{blue}{\left(y \cdot x\right)} \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*l* [=>] 6.5%	\[ \frac{\color{blue}{y \cdot \left(x \cdot z\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 6.6%	\[ \color{blue}{y \cdot \frac{x \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]

   3. Taylor expanded in z around -inf 100.0%

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

   4. Simplified 100.0%

      \[\leadsto y \cdot \color{blue}{\left(-x\right)} \]
      Step-by-step derivation

      [Start] 100.0%	\[ y \cdot \left(-1 \cdot x\right) \]
      neg-mul-1 [<=] 100.0%	\[ y \cdot \color{blue}{\left(-x\right)} \]

   if -2.99999999999999993e147 < z < -6e-151 or 2.10000000000000003e-207 < z < 6.1e97

   1. Initial program 85.9%

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

   2. Simplified 90.4%

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

      [Start] 85.9%	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-/l* [=>] 90.4%	\[ \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Applied egg-rr 92.1%

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

      [Start] 90.4%	\[ \frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]
      div-inv [=>] 90.4%	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{1}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]
      clear-num [<=] 90.4%	\[ \left(x \cdot y\right) \cdot \color{blue}{\frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]
      associate-*l* [=>] 92.1%	\[ \color{blue}{x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)} \]

   if -6e-151 < z < 2.10000000000000003e-207

   1. Initial program 79.1%

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

   2. Simplified 67.9%

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

      [Start] 79.1%	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-/l* [=>] 67.9%	\[ \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Applied egg-rr 79.2%

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

      [Start] 67.9%	\[ \frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]
      associate-/l* [<=] 79.1%	\[ \color{blue}{\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]
      clear-num [=>] 79.2%	\[ \color{blue}{\frac{1}{\frac{\sqrt{z \cdot z - t \cdot a}}{\left(x \cdot y\right) \cdot z}}} \]
      associate-/r/ [=>] 79.1%	\[ \color{blue}{\frac{1}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(\left(x \cdot y\right) \cdot z\right)} \]
      pow1/2 [=>] 79.1%	\[ \frac{1}{\color{blue}{{\left(z \cdot z - t \cdot a\right)}^{0.5}}} \cdot \left(\left(x \cdot y\right) \cdot z\right) \]
      pow-flip [=>] 79.2%	\[ \color{blue}{{\left(z \cdot z - t \cdot a\right)}^{\left(-0.5\right)}} \cdot \left(\left(x \cdot y\right) \cdot z\right) \]
      metadata-eval [=>] 79.2%	\[ {\left(z \cdot z - t \cdot a\right)}^{\color{blue}{-0.5}} \cdot \left(\left(x \cdot y\right) \cdot z\right) \]
      *-commutative [=>] 79.2%	\[ {\left(z \cdot z - t \cdot a\right)}^{-0.5} \cdot \color{blue}{\left(z \cdot \left(x \cdot y\right)\right)} \]

   4. Taylor expanded in z around 0 79.2%

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

   5. Simplified 79.2%

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

      [Start] 79.2%	\[ {\left(-1 \cdot \left(a \cdot t\right)\right)}^{-0.5} \cdot \left(z \cdot \left(x \cdot y\right)\right) \]
      associate-*r* [=>] 79.2%	\[ {\color{blue}{\left(\left(-1 \cdot a\right) \cdot t\right)}}^{-0.5} \cdot \left(z \cdot \left(x \cdot y\right)\right) \]
      neg-mul-1 [<=] 79.2%	\[ {\left(\color{blue}{\left(-a\right)} \cdot t\right)}^{-0.5} \cdot \left(z \cdot \left(x \cdot y\right)\right) \]

   if 6.1e97 < z

   1. Initial program 37.2%

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

   2. Simplified 39.1%

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

      [Start] 37.2%	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-/l* [=>] 39.1%	\[ \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Taylor expanded in z around inf 92.4%

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

   4. Applied egg-rr 84.1%

      \[\leadsto \frac{x \cdot y}{\frac{z + -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{a}{\frac{z}{t}}\right)} - 1\right)}}{z}} \]
      Step-by-step derivation

      [Start] 92.4%	\[ \frac{x \cdot y}{\frac{z + -0.5 \cdot \frac{a \cdot t}{z}}{z}} \]
      expm1-log1p-u [=>] 82.6%	\[ \frac{x \cdot y}{\frac{z + -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot t}{z}\right)\right)}}{z}} \]
      expm1-udef [=>] 82.6%	\[ \frac{x \cdot y}{\frac{z + -0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{a \cdot t}{z}\right)} - 1\right)}}{z}} \]
      associate-/l* [=>] 84.1%	\[ \frac{x \cdot y}{\frac{z + -0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{\frac{z}{t}}}\right)} - 1\right)}{z}} \]

   5. Simplified 98.6%

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

      [Start] 84.1%	\[ \frac{x \cdot y}{\frac{z + -0.5 \cdot \left(e^{\mathsf{log1p}\left(\frac{a}{\frac{z}{t}}\right)} - 1\right)}{z}} \]
      expm1-def [=>] 84.1%	\[ \frac{x \cdot y}{\frac{z + -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\frac{z}{t}}\right)\right)}}{z}} \]
      expm1-log1p [=>] 98.6%	\[ \frac{x \cdot y}{\frac{z + -0.5 \cdot \color{blue}{\frac{a}{\frac{z}{t}}}}{z}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+147}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-151}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-207}:\\ \;\;\;\;{\left(t \cdot \left(-a\right)\right)}^{-0.5} \cdot \left(z \cdot \left(y \cdot x\right)\right)\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	91.2%
Cost	7760

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{if}\;z \leq -3 \cdot 10^{+147}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-207}:\\ \;\;\;\;{\left(t \cdot \left(-a\right)\right)}^{-0.5} \cdot \left(z \cdot \left(y \cdot x\right)\right)\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 2
Accuracy	90.0%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+112}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{+97}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 3
Accuracy	83.0%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-80}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{-131}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 4
Accuracy	83.2%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-93}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-131}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 5
Accuracy	75.4%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-163}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{a} \cdot \frac{x \cdot \left(z \cdot z\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Accuracy	75.7%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-154}:\\ \;\;\;\;-2 \cdot \frac{y}{\frac{a}{\frac{z \cdot x}{\frac{t}{z}}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7
Accuracy	75.7%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-156}:\\ \;\;\;\;\frac{y \cdot x}{\frac{t \cdot -0.5}{z} \cdot \frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8
Accuracy	76.0%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \frac{t \cdot a}{z}}\right)\\ \end{array} \]

Alternative 9
Accuracy	77.2%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\frac{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}{z}}\\ \end{array} \]

Alternative 10
Accuracy	73.9%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-101}:\\ \;\;\;\;x \cdot \left(\left(z \cdot y\right) \cdot \frac{1}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11
Accuracy	74.7%
Cost	776

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-147}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-165}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 12
Accuracy	72.7%
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-303}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 13
Accuracy	42.4%
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023165 
(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)))))