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

Average Accuracy: 61.8% → 91.0%

Time: 32.4s

Precision: binary64

Cost: 7760

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{if}\;z \leq -5 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-198}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{a \cdot \left(-t\right)}}{z \cdot x}}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-1 + \frac{a}{z} \cdot \frac{0.5}{\frac{z}{t}}} \cdot \left(-x\right)\\ \end{array} \]

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) (* a t))))))))
   (if (<= z -5e+140)
     (* x (- y))
     (if (<= z -4e-188)
       t_1
       (if (<= z 6.8e-198)
         (/ y (/ (sqrt (* a (- t))) (* z x)))
         (if (<= z 4.3e+135)
           t_1
           (* (/ y (+ -1.0 (* (/ a z) (/ 0.5 (/ z t))))) (- 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) - (a * t)))));
	double tmp;
	if (z <= -5e+140) {
		tmp = x * -y;
	} else if (z <= -4e-188) {
		tmp = t_1;
	} else if (z <= 6.8e-198) {
		tmp = y / (sqrt((a * -t)) / (z * x));
	} else if (z <= 4.3e+135) {
		tmp = t_1;
	} else {
		tmp = (y / (-1.0 + ((a / z) * (0.5 / (z / t))))) * -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) - (a * t)))))
    if (z <= (-5d+140)) then
        tmp = x * -y
    else if (z <= (-4d-188)) then
        tmp = t_1
    else if (z <= 6.8d-198) then
        tmp = y / (sqrt((a * -t)) / (z * x))
    else if (z <= 4.3d+135) then
        tmp = t_1
    else
        tmp = (y / ((-1.0d0) + ((a / z) * (0.5d0 / (z / t))))) * -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) - (a * t)))));
	double tmp;
	if (z <= -5e+140) {
		tmp = x * -y;
	} else if (z <= -4e-188) {
		tmp = t_1;
	} else if (z <= 6.8e-198) {
		tmp = y / (Math.sqrt((a * -t)) / (z * x));
	} else if (z <= 4.3e+135) {
		tmp = t_1;
	} else {
		tmp = (y / (-1.0 + ((a / z) * (0.5 / (z / t))))) * -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) - (a * t)))))
	tmp = 0
	if z <= -5e+140:
		tmp = x * -y
	elif z <= -4e-188:
		tmp = t_1
	elif z <= 6.8e-198:
		tmp = y / (math.sqrt((a * -t)) / (z * x))
	elif z <= 4.3e+135:
		tmp = t_1
	else:
		tmp = (y / (-1.0 + ((a / z) * (0.5 / (z / t))))) * -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(x * Float64(y * Float64(z / sqrt(Float64(Float64(z * z) - Float64(a * t))))))
	tmp = 0.0
	if (z <= -5e+140)
		tmp = Float64(x * Float64(-y));
	elseif (z <= -4e-188)
		tmp = t_1;
	elseif (z <= 6.8e-198)
		tmp = Float64(y / Float64(sqrt(Float64(a * Float64(-t))) / Float64(z * x)));
	elseif (z <= 4.3e+135)
		tmp = t_1;
	else
		tmp = Float64(Float64(y / Float64(-1.0 + Float64(Float64(a / z) * Float64(0.5 / Float64(z / t))))) * Float64(-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) - (a * t)))));
	tmp = 0.0;
	if (z <= -5e+140)
		tmp = x * -y;
	elseif (z <= -4e-188)
		tmp = t_1;
	elseif (z <= 6.8e-198)
		tmp = y / (sqrt((a * -t)) / (z * x));
	elseif (z <= 4.3e+135)
		tmp = t_1;
	else
		tmp = (y / (-1.0 + ((a / z) * (0.5 / (z / t))))) * -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[(x * N[(y * N[(z / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5e+140], N[(x * (-y)), $MachinePrecision], If[LessEqual[z, -4e-188], t$95$1, If[LessEqual[z, 6.8e-198], N[(y / N[(N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.3e+135], t$95$1, N[(N[(y / N[(-1.0 + N[(N[(a / z), $MachinePrecision] * N[(0.5 / N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]]]]]]

Target

Original	61.8%
Target	88.4%
Herbie	91.0%

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

Derivation

1. Split input into 4 regimes

2. if z < -5.00000000000000008e140

   1. Initial program 20.2%

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

   2. Simplified 21.6%

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

      [Start] 20.2	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 21.6	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]
      associate-*l* [=>] 21.6	\[ \color{blue}{x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)} \]

   3. Taylor expanded in z around -inf 97.8%

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

   4. Simplified 97.8%

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

      [Start] 97.8	\[ -1 \cdot \left(y \cdot x\right) \]
      associate-*r* [=>] 97.8	\[ \color{blue}{\left(-1 \cdot y\right) \cdot x} \]
      mul-1-neg [=>] 97.8	\[ \color{blue}{\left(-y\right)} \cdot x \]

   if -5.00000000000000008e140 < z < -3.9999999999999998e-188 or 6.7999999999999996e-198 < z < 4.29999999999999972e135

   1. Initial program 86.8%

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

   2. Simplified 91.0%

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

      [Start] 86.8	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 91.4	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]
      associate-*l* [=>] 91.0	\[ \color{blue}{x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)} \]

   if -3.9999999999999998e-188 < z < 6.7999999999999996e-198

   1. Initial program 71.4%

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

   2. Simplified 71.2%

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

      [Start] 71.4	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*l* [=>] 71.2	\[ \frac{\color{blue}{x \cdot \left(y \cdot z\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-/l* [=>] 71.2	\[ \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{y \cdot z}}} \]

   3. Applied egg-rr 70.7%

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

      [Start] 71.2	\[ \frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{y \cdot z}} \]
      associate-/l/ [<=] 68.2	\[ \frac{x}{\color{blue}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{y}}} \]
      associate-/r/ [=>] 70.7	\[ \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot y} \]

   4. Applied egg-rr 73.8%

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

      [Start] 70.7	\[ \frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot y \]
      *-commutative [=>] 70.7	\[ \color{blue}{y \cdot \frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]
      clear-num [=>] 70.3	\[ y \cdot \color{blue}{\frac{1}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{x}}} \]
      un-div-inv [=>] 70.3	\[ \color{blue}{\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}{x}}} \]
      associate-/l/ [=>] 73.8	\[ \frac{y}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

   5. Taylor expanded in z around 0 73.8%

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

   6. Simplified 73.8%

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

      [Start] 73.8	\[ \frac{y}{\frac{\sqrt{-1 \cdot \left(a \cdot t\right)}}{x \cdot z}} \]
      associate-*r* [=>] 73.8	\[ \frac{y}{\frac{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot t}}}{x \cdot z}} \]
      neg-mul-1 [<=] 73.8	\[ \frac{y}{\frac{\sqrt{\color{blue}{\left(-a\right)} \cdot t}}{x \cdot z}} \]
      *-commutative [<=] 73.8	\[ \frac{y}{\frac{\sqrt{\color{blue}{t \cdot \left(-a\right)}}}{x \cdot z}} \]

   if 4.29999999999999972e135 < z

   1. Initial program 21.0%

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

   2. Simplified 21.0%

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

      [Start] 21.0	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*l* [=>] 19.6	\[ \frac{\color{blue}{x \cdot \left(y \cdot z\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-/l* [=>] 21.0	\[ \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{y \cdot z}}} \]

   3. Taylor expanded in z around inf 89.3%

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

   4. Simplified 97.2%

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

      [Start] 89.3	\[ \frac{x}{\frac{1}{y} + -0.5 \cdot \frac{a \cdot t}{y \cdot {z}^{2}}} \]
      associate-/l* [=>] 97.2	\[ \frac{x}{\frac{1}{y} + -0.5 \cdot \color{blue}{\frac{a}{\frac{y \cdot {z}^{2}}{t}}}} \]
      unpow2 [=>] 97.2	\[ \frac{x}{\frac{1}{y} + -0.5 \cdot \frac{a}{\frac{y \cdot \color{blue}{\left(z \cdot z\right)}}{t}}} \]

   5. Taylor expanded in y around 0 89.7%

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

   6. Simplified 97.4%

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

      [Start] 89.7	\[ \frac{y \cdot x}{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}} \]
      associate-/l* [=>] 89.4	\[ \color{blue}{\frac{y}{\frac{1 + -0.5 \cdot \frac{a \cdot t}{{z}^{2}}}{x}}} \]
      associate-/l* [=>] 97.2	\[ \frac{y}{\frac{1 + -0.5 \cdot \color{blue}{\frac{a}{\frac{{z}^{2}}{t}}}}{x}} \]
      unpow2 [=>] 97.2	\[ \frac{y}{\frac{1 + -0.5 \cdot \frac{a}{\frac{\color{blue}{z \cdot z}}{t}}}{x}} \]
      associate-/l* [=>] 97.4	\[ \frac{y}{\frac{1 + -0.5 \cdot \frac{a}{\color{blue}{\frac{z}{\frac{t}{z}}}}}{x}} \]

   7. Applied egg-rr 97.9%

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

      [Start] 97.4	\[ \frac{y}{\frac{1 + -0.5 \cdot \frac{a}{\frac{z}{\frac{t}{z}}}}{x}} \]
      frac-2neg [=>] 97.4	\[ \frac{y}{\color{blue}{\frac{-\left(1 + -0.5 \cdot \frac{a}{\frac{z}{\frac{t}{z}}}\right)}{-x}}} \]
      associate-/r/ [=>] 97.8	\[ \color{blue}{\frac{y}{-\left(1 + -0.5 \cdot \frac{a}{\frac{z}{\frac{t}{z}}}\right)} \cdot \left(-x\right)} \]
      neg-sub0 [=>] 97.8	\[ \frac{y}{\color{blue}{0 - \left(1 + -0.5 \cdot \frac{a}{\frac{z}{\frac{t}{z}}}\right)}} \cdot \left(-x\right) \]
      metadata-eval [<=] 97.8	\[ \frac{y}{\color{blue}{\log 1} - \left(1 + -0.5 \cdot \frac{a}{\frac{z}{\frac{t}{z}}}\right)} \cdot \left(-x\right) \]
      associate--r+ [=>] 97.8	\[ \frac{y}{\color{blue}{\left(\log 1 - 1\right) - -0.5 \cdot \frac{a}{\frac{z}{\frac{t}{z}}}}} \cdot \left(-x\right) \]
      metadata-eval [=>] 97.8	\[ \frac{y}{\left(\color{blue}{0} - 1\right) - -0.5 \cdot \frac{a}{\frac{z}{\frac{t}{z}}}} \cdot \left(-x\right) \]
      metadata-eval [=>] 97.8	\[ \frac{y}{\color{blue}{-1} - -0.5 \cdot \frac{a}{\frac{z}{\frac{t}{z}}}} \cdot \left(-x\right) \]
      associate-*r/ [=>] 97.8	\[ \frac{y}{-1 - \color{blue}{\frac{-0.5 \cdot a}{\frac{z}{\frac{t}{z}}}}} \cdot \left(-x\right) \]
      associate-/r/ [=>] 97.8	\[ \frac{y}{-1 - \frac{-0.5 \cdot a}{\color{blue}{\frac{z}{t} \cdot z}}} \cdot \left(-x\right) \]
      times-frac [=>] 97.9	\[ \frac{y}{-1 - \color{blue}{\frac{-0.5}{\frac{z}{t}} \cdot \frac{a}{z}}} \cdot \left(-x\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-188}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-198}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{a \cdot \left(-t\right)}}{z \cdot x}}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+135}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-1 + \frac{a}{z} \cdot \frac{0.5}{\frac{z}{t}}} \cdot \left(-x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	81.5%
Cost	7568

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot \frac{z}{\sqrt{a \cdot \left(-t\right)}}\right)\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{0.5 \cdot \frac{a}{\frac{y \cdot \left(z \cdot z\right)}{t}} + \frac{-1}{y}}\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-137}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \frac{a \cdot t}{z} - z}{z}}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-171}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-1 + \frac{a}{z} \cdot \frac{0.5}{\frac{z}{t}}} \cdot \left(-x\right)\\ \end{array} \]

Alternative 2
Accuracy	82.0%
Cost	7568

\[\begin{array}{l} t_1 := \sqrt{a \cdot \left(-t\right)}\\ \mathbf{if}\;z \leq -2.85 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{0.5 \cdot \frac{a}{\frac{y \cdot \left(z \cdot z\right)}{t}} + \frac{-1}{y}}\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-109}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{t_1}\right)\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-138}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \frac{a \cdot t}{z} - z}{z}}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-171}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-1 + \frac{a}{z} \cdot \frac{0.5}{\frac{z}{t}}} \cdot \left(-x\right)\\ \end{array} \]

Alternative 3
Accuracy	81.6%
Cost	7568

\[\begin{array}{l} t_1 := \sqrt{a \cdot \left(-t\right)}\\ \mathbf{if}\;z \leq -5.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{0.5 \cdot \frac{a}{\frac{y \cdot \left(z \cdot z\right)}{t}} + \frac{-1}{y}}\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-108}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{t_1}\right)\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-144}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \frac{a \cdot t}{z} - z}{z}}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-168}:\\ \;\;\;\;y \cdot \frac{x}{\frac{t_1}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-1 + \frac{a}{z} \cdot \frac{0.5}{\frac{z}{t}}} \cdot \left(-x\right)\\ \end{array} \]

Alternative 4
Accuracy	90.9%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+140}:\\ \;\;\;\;\frac{x \cdot y}{\frac{\mathsf{fma}\left(0.5, \frac{a}{\frac{z}{t}}, -z\right)}{z}}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+135}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-1 + \frac{a}{z} \cdot \frac{0.5}{\frac{z}{t}}} \cdot \left(-x\right)\\ \end{array} \]

Alternative 5
Accuracy	90.9%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+135}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\sqrt{z \cdot z - a \cdot t}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-1 + \frac{a}{z} \cdot \frac{0.5}{\frac{z}{t}}} \cdot \left(-x\right)\\ \end{array} \]

Alternative 6
Accuracy	76.0%
Cost	1156

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-233}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-1 + \frac{a}{z} \cdot \frac{0.5}{\frac{z}{t}}} \cdot \left(-x\right)\\ \end{array} \]

Alternative 7
Accuracy	77.1%
Cost	1156

\[\begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-255}:\\ \;\;\;\;y \cdot \frac{x}{\frac{0.5 \cdot \frac{a \cdot t}{z} - z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{-1 + \frac{a}{z} \cdot \frac{0.5}{\frac{z}{t}}} \cdot \left(-x\right)\\ \end{array} \]

Alternative 8
Accuracy	74.8%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-233}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \frac{a \cdot t}{z}}\right)\\ \end{array} \]

Alternative 9
Accuracy	75.9%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-235}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z + \frac{a}{\frac{z}{t}} \cdot -0.5} \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 10
Accuracy	73.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-233}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-107}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11
Accuracy	70.8%
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-305}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 12
Accuracy	42.5%
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2023137 
(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)))))