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

Average Error: 24.8 → 6.5

Time: 38.3s

Precision: binary64

Cost: 7496

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+103}:\\ \;\;\;\;\frac{z}{\frac{0.5 \cdot t}{\frac{z}{a}} - z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - t \cdot \frac{a}{z \cdot z}}}\\ \end{array} \]

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
 (if (<= z -6.8e+103)
   (* (/ z (- (/ (* 0.5 t) (/ z a)) z)) (* x y))
   (if (<= z 5.5e-27)
     (/ x (/ (sqrt (- (* z z) (* t a))) (* z y)))
     (/ (* x y) (sqrt (- 1.0 (* t (/ a (* z 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 tmp;
	if (z <= -6.8e+103) {
		tmp = (z / (((0.5 * t) / (z / a)) - z)) * (x * y);
	} else if (z <= 5.5e-27) {
		tmp = x / (sqrt(((z * z) - (t * a))) / (z * y));
	} else {
		tmp = (x * y) / sqrt((1.0 - (t * (a / (z * 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) :: tmp
    if (z <= (-6.8d+103)) then
        tmp = (z / (((0.5d0 * t) / (z / a)) - z)) * (x * y)
    else if (z <= 5.5d-27) then
        tmp = x / (sqrt(((z * z) - (t * a))) / (z * y))
    else
        tmp = (x * y) / sqrt((1.0d0 - (t * (a / (z * 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 tmp;
	if (z <= -6.8e+103) {
		tmp = (z / (((0.5 * t) / (z / a)) - z)) * (x * y);
	} else if (z <= 5.5e-27) {
		tmp = x / (Math.sqrt(((z * z) - (t * a))) / (z * y));
	} else {
		tmp = (x * y) / Math.sqrt((1.0 - (t * (a / (z * 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):
	tmp = 0
	if z <= -6.8e+103:
		tmp = (z / (((0.5 * t) / (z / a)) - z)) * (x * y)
	elif z <= 5.5e-27:
		tmp = x / (math.sqrt(((z * z) - (t * a))) / (z * y))
	else:
		tmp = (x * y) / math.sqrt((1.0 - (t * (a / (z * 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)
	tmp = 0.0
	if (z <= -6.8e+103)
		tmp = Float64(Float64(z / Float64(Float64(Float64(0.5 * t) / Float64(z / a)) - z)) * Float64(x * y));
	elseif (z <= 5.5e-27)
		tmp = Float64(x / Float64(sqrt(Float64(Float64(z * z) - Float64(t * a))) / Float64(z * y)));
	else
		tmp = Float64(Float64(x * y) / sqrt(Float64(1.0 - Float64(t * Float64(a / Float64(z * 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)
	tmp = 0.0;
	if (z <= -6.8e+103)
		tmp = (z / (((0.5 * t) / (z / a)) - z)) * (x * y);
	elseif (z <= 5.5e-27)
		tmp = x / (sqrt(((z * z) - (t * a))) / (z * y));
	else
		tmp = (x * y) / sqrt((1.0 - (t * (a / (z * 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_] := If[LessEqual[z, -6.8e+103], N[(N[(z / N[(N[(N[(0.5 * t), $MachinePrecision] / N[(z / a), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.5e-27], N[(x / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t * N[(a / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	24.8
Target	7.6
Herbie	6.5

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

Derivation

1. Split input into 3 regimes

2. if z < -6.7999999999999997e103

   1. Initial program 44.4

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

   2. Simplified 41.6

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

      [Start] 44.4	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 41.6	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]
      *-commutative [<=] 41.6	\[ \color{blue}{\frac{z}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(x \cdot y\right)} \]

   3. Taylor expanded in z around -inf 5.5

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

   4. Applied egg-rr 7.0

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

   5. Simplified 1.8

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

      [Start] 7.0	\[ \frac{z}{\left(e^{\mathsf{log1p}\left(\frac{0.5}{\frac{z}{a}} \cdot t\right)} - 1\right) + -1 \cdot z} \cdot \left(x \cdot y\right) \]
      expm1-def [=>] 7.0	\[ \frac{z}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{\frac{z}{a}} \cdot t\right)\right)} + -1 \cdot z} \cdot \left(x \cdot y\right) \]
      expm1-log1p [=>] 1.8	\[ \frac{z}{\color{blue}{\frac{0.5}{\frac{z}{a}} \cdot t} + -1 \cdot z} \cdot \left(x \cdot y\right) \]
      associate-*l/ [=>] 1.8	\[ \frac{z}{\color{blue}{\frac{0.5 \cdot t}{\frac{z}{a}}} + -1 \cdot z} \cdot \left(x \cdot y\right) \]

   if -6.7999999999999997e103 < z < 5.5000000000000002e-27

   1. Initial program 12.6

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

   2. Simplified 12.0

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

      [Start] 12.6	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*l* [=>] 12.9	\[ \frac{\color{blue}{x \cdot \left(y \cdot z\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-/l* [=>] 12.0	\[ \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{y \cdot z}}} \]

   if 5.5000000000000002e-27 < z

   1. Initial program 30.1

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

   2. Simplified 27.7

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

      [Start] 30.1	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-/l* [=>] 27.7	\[ \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Applied egg-rr 33.8

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

   4. Simplified 1.4

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

      [Start] 33.8	\[ \frac{x \cdot y}{\sqrt{\frac{z \cdot z - t \cdot a}{z \cdot z}}} \]
      *-commutative [=>] 33.8	\[ \frac{x \cdot y}{\sqrt{\frac{z \cdot z - \color{blue}{a \cdot t}}{z \cdot z}}} \]
      div-sub [=>] 33.8	\[ \frac{x \cdot y}{\sqrt{\color{blue}{\frac{z \cdot z}{z \cdot z} - \frac{a \cdot t}{z \cdot z}}}} \]
      unpow2 [<=] 33.8	\[ \frac{x \cdot y}{\sqrt{\frac{\color{blue}{{z}^{2}}}{z \cdot z} - \frac{a \cdot t}{z \cdot z}}} \]
      unpow2 [<=] 33.8	\[ \frac{x \cdot y}{\sqrt{\frac{{z}^{2}}{\color{blue}{{z}^{2}}} - \frac{a \cdot t}{z \cdot z}}} \]
      *-inverses [=>] 5.0	\[ \frac{x \cdot y}{\sqrt{\color{blue}{1} - \frac{a \cdot t}{z \cdot z}}} \]
      unpow2 [<=] 5.0	\[ \frac{x \cdot y}{\sqrt{1 - \frac{a \cdot t}{\color{blue}{{z}^{2}}}}} \]
      associate-/l* [=>] 1.4	\[ \frac{x \cdot y}{\sqrt{1 - \color{blue}{\frac{a}{\frac{{z}^{2}}{t}}}}} \]
      associate-/r/ [=>] 1.4	\[ \frac{x \cdot y}{\sqrt{1 - \color{blue}{\frac{a}{{z}^{2}} \cdot t}}} \]
      unpow2 [=>] 1.4	\[ \frac{x \cdot y}{\sqrt{1 - \frac{a}{\color{blue}{z \cdot z}} \cdot t}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+103}:\\ \;\;\;\;\frac{z}{\frac{0.5 \cdot t}{\frac{z}{a}} - z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\sqrt{1 - t \cdot \frac{a}{z \cdot z}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.4
Cost	7760

\[\begin{array}{l} t_1 := \frac{y}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(z \cdot x\right)\\ \mathbf{if}\;z \leq -3.7 \cdot 10^{+115}:\\ \;\;\;\;\frac{z}{\frac{0.5 \cdot t}{\frac{z}{a}} - z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-195}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{t \cdot \left(-a\right)}}{z \cdot y}}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \end{array} \]

Alternative 2
Error	7.3
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+121}:\\ \;\;\;\;\frac{z}{\frac{0.5 \cdot t}{\frac{z}{a}} - z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+48}:\\ \;\;\;\;y \cdot \left(z \cdot \frac{x}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \end{array} \]

Alternative 3
Error	7.5
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{z}{\frac{0.5 \cdot t}{\frac{z}{a}} - z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \end{array} \]

Alternative 4
Error	7.4
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+124}:\\ \;\;\;\;\frac{z}{\frac{0.5 \cdot t}{\frac{z}{a}} - z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \frac{z}{\frac{\sqrt{z \cdot z - t \cdot a}}{y}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \end{array} \]

Alternative 5
Error	7.3
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{z}{\frac{0.5 \cdot t}{\frac{z}{a}} - z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \end{array} \]

Alternative 6
Error	11.3
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{z}{\frac{0.5 \cdot t}{\frac{z}{a}} - z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-94}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \end{array} \]

Alternative 7
Error	11.5
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-113}:\\ \;\;\;\;\frac{z}{\frac{0.5 \cdot t}{\frac{z}{a}} - z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-96}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \end{array} \]

Alternative 8
Error	11.4
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-111}:\\ \;\;\;\;\frac{z}{\frac{0.5 \cdot t}{\frac{z}{a}} - z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{t \cdot \left(-a\right)}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \end{array} \]

Alternative 9
Error	15.1
Cost	1288

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+66}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-228}:\\ \;\;\;\;\frac{\left(-z\right) \cdot \left(x \cdot y\right)}{z + \frac{t \cdot -0.5}{\frac{z}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \end{array} \]

Alternative 10
Error	16.4
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-141}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4100:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 11
Error	14.7
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+126}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-288}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{0.5 \cdot \frac{t \cdot a}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \end{array} \]

Alternative 12
Error	15.8
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-143}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \end{array} \]

Alternative 13
Error	14.6
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-288}:\\ \;\;\;\;\frac{z}{\frac{0.5 \cdot t}{\frac{z}{a}} - z} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \end{array} \]

Alternative 14
Error	18.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-170}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 15
Error	17.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-84}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-126}:\\ \;\;\;\;-1 + \left(1 - x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 16
Error	19.3
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 17
Error	36.7
Cost	192

\[x \cdot y \]

Reproduce

herbie shell --seed 2023039 
(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)))))