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

Average Error: 24.6 → 5.5

Time: 32.3s

Precision: binary64

Cost: 13964

\[ \begin{array}{c}[t, a] = \mathsf{sort}([t, a])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot \left(z \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.5}\right)\right)\\ \mathbf{if}\;z \leq -7.107067327257021 \cdot 10^{+138}:\\ \;\;\;\;-0.5 \cdot \left(\frac{y}{\frac{z}{a}} \cdot \frac{t}{\frac{z}{x}}\right) - y \cdot x\\ \mathbf{elif}\;z \leq -9.8 \cdot 10^{-175}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-124}:\\ \;\;\;\;y \cdot \left(\left(z \cdot x\right) \cdot \left({\left(-t\right)}^{-0.5} \cdot {a}^{-0.5}\right)\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

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 (pow (- (* z z) (* a t)) -0.5))))))
   (if (<= z -7.107067327257021e+138)
     (- (* -0.5 (* (/ y (/ z a)) (/ t (/ z x)))) (* y x))
     (if (<= z -9.8e-175)
       t_1
       (if (<= z 1.12e-124)
         (* y (* (* z x) (* (pow (- t) -0.5) (pow a -0.5))))
         (if (<= z 2.3e+45) 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 * pow(((z * z) - (a * t)), -0.5)));
	double tmp;
	if (z <= -7.107067327257021e+138) {
		tmp = (-0.5 * ((y / (z / a)) * (t / (z / x)))) - (y * x);
	} else if (z <= -9.8e-175) {
		tmp = t_1;
	} else if (z <= 1.12e-124) {
		tmp = y * ((z * x) * (pow(-t, -0.5) * pow(a, -0.5)));
	} else if (z <= 2.3e+45) {
		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 * (((z * z) - (a * t)) ** (-0.5d0))))
    if (z <= (-7.107067327257021d+138)) then
        tmp = ((-0.5d0) * ((y / (z / a)) * (t / (z / x)))) - (y * x)
    else if (z <= (-9.8d-175)) then
        tmp = t_1
    else if (z <= 1.12d-124) then
        tmp = y * ((z * x) * ((-t ** (-0.5d0)) * (a ** (-0.5d0))))
    else if (z <= 2.3d+45) 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.pow(((z * z) - (a * t)), -0.5)));
	double tmp;
	if (z <= -7.107067327257021e+138) {
		tmp = (-0.5 * ((y / (z / a)) * (t / (z / x)))) - (y * x);
	} else if (z <= -9.8e-175) {
		tmp = t_1;
	} else if (z <= 1.12e-124) {
		tmp = y * ((z * x) * (Math.pow(-t, -0.5) * Math.pow(a, -0.5)));
	} else if (z <= 2.3e+45) {
		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.pow(((z * z) - (a * t)), -0.5)))
	tmp = 0
	if z <= -7.107067327257021e+138:
		tmp = (-0.5 * ((y / (z / a)) * (t / (z / x)))) - (y * x)
	elif z <= -9.8e-175:
		tmp = t_1
	elif z <= 1.12e-124:
		tmp = y * ((z * x) * (math.pow(-t, -0.5) * math.pow(a, -0.5)))
	elif z <= 2.3e+45:
		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(x * Float64(y * Float64(z * (Float64(Float64(z * z) - Float64(a * t)) ^ -0.5))))
	tmp = 0.0
	if (z <= -7.107067327257021e+138)
		tmp = Float64(Float64(-0.5 * Float64(Float64(y / Float64(z / a)) * Float64(t / Float64(z / x)))) - Float64(y * x));
	elseif (z <= -9.8e-175)
		tmp = t_1;
	elseif (z <= 1.12e-124)
		tmp = Float64(y * Float64(Float64(z * x) * Float64((Float64(-t) ^ -0.5) * (a ^ -0.5))));
	elseif (z <= 2.3e+45)
		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 * (((z * z) - (a * t)) ^ -0.5)));
	tmp = 0.0;
	if (z <= -7.107067327257021e+138)
		tmp = (-0.5 * ((y / (z / a)) * (t / (z / x)))) - (y * x);
	elseif (z <= -9.8e-175)
		tmp = t_1;
	elseif (z <= 1.12e-124)
		tmp = y * ((z * x) * ((-t ^ -0.5) * (a ^ -0.5)));
	elseif (z <= 2.3e+45)
		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[(x * N[(y * N[(z * N[Power[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.107067327257021e+138], N[(N[(-0.5 * N[(N[(y / N[(z / a), $MachinePrecision]), $MachinePrecision] * N[(t / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -9.8e-175], t$95$1, If[LessEqual[z, 1.12e-124], N[(y * N[(N[(z * x), $MachinePrecision] * N[(N[Power[(-t), -0.5], $MachinePrecision] * N[Power[a, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.3e+45], t$95$1, N[(y * x), $MachinePrecision]]]]]]

Target

Original	24.6
Target	7.5
Herbie	5.5

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

Derivation

1. Split input into 4 regimes

2. if z < -7.10706732725702149e138

   1. Initial program 50.6

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

   2. Applied egg-rr 50.0

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

   3. Taylor expanded in x around 0 51.6

      \[\leadsto \color{blue}{\sqrt{\frac{1}{{z}^{2} - a \cdot t}} \cdot \left(y \cdot \left(z \cdot x\right)\right)} \]

   4. Simplified 50.9

      \[\leadsto \color{blue}{y \cdot \left(\left(z \cdot x\right) \cdot \sqrt{\frac{1}{z \cdot z - a \cdot t}}\right)} \]
      Proof
      (*.f64 y (*.f64 (*.f64 z x) (sqrt.f64 (/.f64 1 (-.f64 (*.f64 z z) (*.f64 a t)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 y (*.f64 (*.f64 z x) (sqrt.f64 (/.f64 1 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)) (*.f64 a t)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y (*.f64 z x)) (sqrt.f64 (/.f64 1 (-.f64 (pow.f64 z 2) (*.f64 a t)))))): 44 points increase in error, 11 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 (/.f64 1 (-.f64 (pow.f64 z 2) (*.f64 a t)))) (*.f64 y (*.f64 z x)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in z around -inf 13.5

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

   6. Simplified 1.8

      \[\leadsto \color{blue}{-0.5 \cdot \left(\frac{y}{\frac{z}{a}} \cdot \frac{t}{\frac{z}{x}}\right) - x \cdot y} \]
      Proof
      (-.f64 (*.f64 -1/2 (*.f64 (/.f64 y (/.f64 z a)) (/.f64 t (/.f64 z x)))) (*.f64 x y)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1/2 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y a) z)) (/.f64 t (/.f64 z x)))) (*.f64 x y)): 17 points increase in error, 5 points decrease in error
      (-.f64 (*.f64 -1/2 (*.f64 (/.f64 (*.f64 y a) z) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t x) z)))) (*.f64 x y)): 13 points increase in error, 8 points decrease in error
      (-.f64 (*.f64 -1/2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 y a) (*.f64 t x)) (*.f64 z z)))) (*.f64 x y)): 17 points increase in error, 8 points decrease in error
      (-.f64 (*.f64 -1/2 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 y (*.f64 a (*.f64 t x)))) (*.f64 z z))) (*.f64 x y)): 8 points increase in error, 9 points decrease in error
      (-.f64 (*.f64 -1/2 (/.f64 (*.f64 y (*.f64 a (*.f64 t x))) (Rewrite<= unpow2_binary64 (pow.f64 z 2)))) (*.f64 x y)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1/2 (/.f64 (*.f64 y (*.f64 a (*.f64 t x))) (pow.f64 z 2))) (Rewrite<= *-commutative_binary64 (*.f64 y x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1/2 (/.f64 (*.f64 y (*.f64 a (*.f64 t x))) (pow.f64 z 2))) (neg.f64 (*.f64 y x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (/.f64 (*.f64 y (*.f64 a (*.f64 t x))) (pow.f64 z 2))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 y x)))): 0 points increase in error, 0 points decrease in error

   if -7.10706732725702149e138 < z < -9.79999999999999996e-175 or 1.12e-124 < z < 2.30000000000000012e45

   1. Initial program 8.0

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

   2. Simplified 7.9

      \[\leadsto \color{blue}{x \cdot \frac{y \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]
      Proof
      (*.f64 x (/.f64 (*.f64 y z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x (*.f64 y z)) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))): 34 points increase in error, 10 points decrease in error
      (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x y) z)) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))): 16 points increase in error, 42 points decrease in error

   3. Applied egg-rr 4.8

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

   if -9.79999999999999996e-175 < z < 1.12e-124

   1. Initial program 16.9

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

   2. Taylor expanded in x around 0 15.5

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

   3. Applied egg-rr 15.1

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

   4. Taylor expanded in a around inf 13.6

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

   5. Simplified 11.5

      \[\leadsto y \cdot \left(\left(z \cdot x\right) \cdot \color{blue}{\left({\left(-t\right)}^{-0.5} \cdot {a}^{-0.5}\right)}\right) \]
      Proof
      (*.f64 (pow.f64 (neg.f64 t) -1/2) (pow.f64 a -1/2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (neg.f64 t)) -1/2))) (pow.f64 a -1/2)): 84 points increase in error, 82 points decrease in error
      (*.f64 (exp.f64 (*.f64 (log.f64 (neg.f64 t)) -1/2)) (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 a) -1/2)))): 67 points increase in error, 100 points decrease in error
      (*.f64 (exp.f64 (*.f64 (log.f64 (neg.f64 t)) -1/2)) (exp.f64 (*.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (log.f64 a)))) -1/2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (*.f64 (log.f64 (neg.f64 t)) -1/2)) (exp.f64 (*.f64 (neg.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 a)))) -1/2))): 0 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (*.f64 (log.f64 (neg.f64 t)) -1/2)) (exp.f64 (*.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 (/.f64 1 a)))) -1/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= exp-sum_binary64 (exp.f64 (+.f64 (*.f64 (log.f64 (neg.f64 t)) -1/2) (*.f64 (*.f64 -1 (log.f64 (/.f64 1 a))) -1/2)))): 58 points increase in error, 52 points decrease in error
      (exp.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 -1/2 (+.f64 (log.f64 (neg.f64 t)) (*.f64 -1 (log.f64 (/.f64 1 a))))))): 0 points increase in error, 0 points decrease in error

   if 2.30000000000000012e45 < z

   1. Initial program 36.9

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

   2. Taylor expanded in z around inf 4.5

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

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.107067327257021 \cdot 10^{+138}:\\ \;\;\;\;-0.5 \cdot \left(\frac{y}{\frac{z}{a}} \cdot \frac{t}{\frac{z}{x}}\right) - y \cdot x\\ \mathbf{elif}\;z \leq -9.8 \cdot 10^{-175}:\\ \;\;\;\;x \cdot \left(y \cdot \left(z \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.5}\right)\right)\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-124}:\\ \;\;\;\;y \cdot \left(\left(z \cdot x\right) \cdot \left({\left(-t\right)}^{-0.5} \cdot {a}^{-0.5}\right)\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+45}:\\ \;\;\;\;x \cdot \left(y \cdot \left(z \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Error	8.0
Cost	7760

\[\begin{array}{l} t_1 := \frac{x \cdot \left(z \cdot y\right)}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+32}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-175}:\\ \;\;\;\;z \cdot \left(\left(y \cdot x\right) \cdot {\left(a \cdot \left(-t\right)\right)}^{-0.5}\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	7.0
Cost	7560

\[\begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+40}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.42 \cdot 10^{+87}:\\ \;\;\;\;y \cdot \left({\left(z \cdot z - a \cdot t\right)}^{-0.5} \cdot \left(z \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Error	6.7
Cost	7560

\[\begin{array}{l} \mathbf{if}\;z \leq -7.107067327257021 \cdot 10^{+138}:\\ \;\;\;\;-0.5 \cdot \left(\frac{y}{\frac{z}{a}} \cdot \frac{t}{\frac{z}{x}}\right) - y \cdot x\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+45}:\\ \;\;\;\;x \cdot \left(y \cdot \left(z \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4
Error	7.7
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+40}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+45}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5
Error	12.8
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Error	16.2
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z + \frac{-0.5}{\frac{z}{a \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7
Error	16.0
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8
Error	17.0
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-161}:\\ \;\;\;\;-2 \cdot \left(\left(x \cdot \left(z \cdot z\right)\right) \cdot \frac{y}{a \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 9
Error	16.7
Cost	1032

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-175}:\\ \;\;\;\;\frac{\left(y \cdot \left(z \cdot x\right) + 1\right) + -1}{-z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10
Error	17.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \left(\left(y + 1\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11
Error	17.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-63}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 12
Error	19.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{-307}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 13
Error	37.1
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2022291 
(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)))))