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

Average Error: 25.2 → 5.2

Time: 52.3s

Precision: binary64

Cost: 13964

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

Math

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

↓

\[\begin{array}{l} t_1 := z \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.5}\\ \mathbf{if}\;z \leq -5.864915385361467 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{\frac{z}{t \cdot 0.5}} - z}{z}} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-180}:\\ \;\;\;\;x \cdot \left(y \cdot t_1\right)\\ \mathbf{elif}\;z \leq 10^{-160}:\\ \;\;\;\;z \cdot \left(\left({\left(-t\right)}^{-0.5} \cdot {a}^{-0.5}\right) \cdot \left(x \cdot y\right)\right)\\ \mathbf{elif}\;z \leq 1.5551926624209129 \cdot 10^{+156}:\\ \;\;\;\;t_1 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

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 (* z (pow (- (* z z) (* a t)) -0.5))))
   (if (<= z -5.864915385361467e+154)
     (* (/ 1.0 (/ (- (/ a (/ z (* t 0.5))) z) z)) (* x y))
     (if (<= z -1e-180)
       (* x (* y t_1))
       (if (<= z 1e-160)
         (* z (* (* (pow (- t) -0.5) (pow a -0.5)) (* x y)))
         (if (<= z 1.5551926624209129e+156) (* t_1 (* x y)) (* x y)))))))

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 = z * pow(((z * z) - (a * t)), -0.5);
	double tmp;
	if (z <= -5.864915385361467e+154) {
		tmp = (1.0 / (((a / (z / (t * 0.5))) - z) / z)) * (x * y);
	} else if (z <= -1e-180) {
		tmp = x * (y * t_1);
	} else if (z <= 1e-160) {
		tmp = z * ((pow(-t, -0.5) * pow(a, -0.5)) * (x * y));
	} else if (z <= 1.5551926624209129e+156) {
		tmp = t_1 * (x * y);
	} else {
		tmp = x * y;
	}
	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 = z * (((z * z) - (a * t)) ** (-0.5d0))
    if (z <= (-5.864915385361467d+154)) then
        tmp = (1.0d0 / (((a / (z / (t * 0.5d0))) - z) / z)) * (x * y)
    else if (z <= (-1d-180)) then
        tmp = x * (y * t_1)
    else if (z <= 1d-160) then
        tmp = z * (((-t ** (-0.5d0)) * (a ** (-0.5d0))) * (x * y))
    else if (z <= 1.5551926624209129d+156) then
        tmp = t_1 * (x * y)
    else
        tmp = x * y
    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 = z * Math.pow(((z * z) - (a * t)), -0.5);
	double tmp;
	if (z <= -5.864915385361467e+154) {
		tmp = (1.0 / (((a / (z / (t * 0.5))) - z) / z)) * (x * y);
	} else if (z <= -1e-180) {
		tmp = x * (y * t_1);
	} else if (z <= 1e-160) {
		tmp = z * ((Math.pow(-t, -0.5) * Math.pow(a, -0.5)) * (x * y));
	} else if (z <= 1.5551926624209129e+156) {
		tmp = t_1 * (x * y);
	} else {
		tmp = x * y;
	}
	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 = z * math.pow(((z * z) - (a * t)), -0.5)
	tmp = 0
	if z <= -5.864915385361467e+154:
		tmp = (1.0 / (((a / (z / (t * 0.5))) - z) / z)) * (x * y)
	elif z <= -1e-180:
		tmp = x * (y * t_1)
	elif z <= 1e-160:
		tmp = z * ((math.pow(-t, -0.5) * math.pow(a, -0.5)) * (x * y))
	elif z <= 1.5551926624209129e+156:
		tmp = t_1 * (x * y)
	else:
		tmp = x * y
	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(z * (Float64(Float64(z * z) - Float64(a * t)) ^ -0.5))
	tmp = 0.0
	if (z <= -5.864915385361467e+154)
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(a / Float64(z / Float64(t * 0.5))) - z) / z)) * Float64(x * y));
	elseif (z <= -1e-180)
		tmp = Float64(x * Float64(y * t_1));
	elseif (z <= 1e-160)
		tmp = Float64(z * Float64(Float64((Float64(-t) ^ -0.5) * (a ^ -0.5)) * Float64(x * y)));
	elseif (z <= 1.5551926624209129e+156)
		tmp = Float64(t_1 * Float64(x * y));
	else
		tmp = Float64(x * y);
	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 = z * (((z * z) - (a * t)) ^ -0.5);
	tmp = 0.0;
	if (z <= -5.864915385361467e+154)
		tmp = (1.0 / (((a / (z / (t * 0.5))) - z) / z)) * (x * y);
	elseif (z <= -1e-180)
		tmp = x * (y * t_1);
	elseif (z <= 1e-160)
		tmp = z * (((-t ^ -0.5) * (a ^ -0.5)) * (x * y));
	elseif (z <= 1.5551926624209129e+156)
		tmp = t_1 * (x * y);
	else
		tmp = x * y;
	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[(z * N[Power[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.864915385361467e+154], N[(N[(1.0 / N[(N[(N[(a / N[(z / N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1e-180], N[(x * N[(y * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e-160], N[(z * N[(N[(N[Power[(-t), -0.5], $MachinePrecision] * N[Power[a, -0.5], $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.5551926624209129e+156], N[(t$95$1 * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]

Target

Original	25.2
Target	7.6
Herbie	5.2

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

Derivation

1. Split input into 5 regimes

2. if z < -5.86491538536146696e154

   1. Initial program 54.1

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

   2. Taylor expanded in z around -inf 21.4

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

   3. Simplified 21.4

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

   4. Applied egg-rr 1.0

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

   if -5.86491538536146696e154 < z < -1e-180

   1. Initial program 9.0

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

   2. Simplified 8.8

      \[\leadsto \color{blue}{x \cdot \frac{y \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]
      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))))): 49 points increase in error, 8 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, 46 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 -1e-180 < z < 9.9999999999999999e-161

   1. Initial program 18.2

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

   2. Applied egg-rr 18.4

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

   3. Taylor expanded in a around inf 15.0

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

   4. Simplified 12.9

      \[\leadsto z \cdot \left(\left(x \cdot y\right) \cdot \color{blue}{\left({\left(-t\right)}^{-0.5} \cdot {a}^{-0.5}\right)}\right) \]
      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)): 78 points increase in error, 90 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)))): 85 points increase in error, 85 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)))): 40 points increase in error, 57 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 9.9999999999999999e-161 < z < 1.5551926624209129e156

   1. Initial program 9.2

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

   2. Applied egg-rr 5.6

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

   if 1.5551926624209129e156 < z

   1. Initial program 54.3

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

   2. Simplified 54.4

      \[\leadsto \color{blue}{x \cdot \frac{y \cdot z}{\sqrt{z \cdot z - t \cdot a}}} \]
      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))))): 49 points increase in error, 8 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, 46 points decrease in error

   3. Taylor expanded in z around inf 1.2

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.864915385361467 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{\frac{z}{t \cdot 0.5}} - z}{z}} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-180}:\\ \;\;\;\;x \cdot \left(y \cdot \left(z \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.5}\right)\right)\\ \mathbf{elif}\;z \leq 10^{-160}:\\ \;\;\;\;z \cdot \left(\left({\left(-t\right)}^{-0.5} \cdot {a}^{-0.5}\right) \cdot \left(x \cdot y\right)\right)\\ \mathbf{elif}\;z \leq 1.5551926624209129 \cdot 10^{+156}:\\ \;\;\;\;\left(z \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.5}\right) \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternatives

Alternative 1
Error	5.7
Cost	13900

\[\begin{array}{l} t_1 := \left(z \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.5}\right) \cdot \left(x \cdot y\right)\\ \mathbf{if}\;z \leq -5.864915385361467 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{\frac{z}{t \cdot 0.5}} - z}{z}} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-255}:\\ \;\;\;\;\frac{x}{\frac{\mathsf{hypot}\left(z, \sqrt{a \cdot \left(-t\right)}\right)}{z \cdot y}}\\ \mathbf{elif}\;z \leq 1.5551926624209129 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 2
Error	6.1
Cost	7560

\[\begin{array}{l} \mathbf{if}\;z \leq -5.864915385361467 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{\frac{z}{t \cdot 0.5}} - z}{z}} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 1.5551926624209129 \cdot 10^{+156}:\\ \;\;\;\;x \cdot \left(y \cdot \left(z \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3
Error	6.4
Cost	7560

\[\begin{array}{l} \mathbf{if}\;z \leq -5.864915385361467 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{\frac{z}{t \cdot 0.5}} - z}{z}} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 1.5551926624209129 \cdot 10^{+156}:\\ \;\;\;\;\left(z \cdot {\left(z \cdot z - a \cdot t\right)}^{-0.5}\right) \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4
Error	8.9
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{\frac{z}{t \cdot 0.5}} - z}{z}} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 32000:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 5
Error	6.9
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{\frac{z}{t \cdot 0.5}} - z}{z}} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 2.06 \cdot 10^{+86}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 6
Error	11.1
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-134}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{\frac{z}{t \cdot 0.5}} - z}{z}} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-89}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7
Error	11.4
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-134}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{\frac{z}{t \cdot 0.5}} - z}{z}} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-89}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 8
Error	16.1
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -1.96 \cdot 10^{-206}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-83}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z + \frac{a \cdot -0.5}{\frac{z}{t}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 9
Error	16.1
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -1.96 \cdot 10^{-206}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-83}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 10
Error	16.1
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -1.96 \cdot 10^{-206}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-83}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11
Error	15.2
Cost	1220

\[\begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{-178}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{\frac{z}{t \cdot 0.5}} - z}{z}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 12
Error	16.8
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -1.96 \cdot 10^{-206}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-157}:\\ \;\;\;\;\frac{-2 \cdot \left(\left(z \cdot y\right) \cdot \left(z \cdot x\right)\right)}{a \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 13
Error	16.5
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -1.96 \cdot 10^{-206}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-178}:\\ \;\;\;\;y \cdot \left(-2 \cdot \frac{z}{a \cdot \frac{t}{z \cdot x}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 14
Error	17.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.96 \cdot 10^{-206}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-175}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 15
Error	17.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.96 \cdot 10^{-206}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-178}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 16
Error	18.9
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-257}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 17
Error	36.8
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2022317 
(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)))))