Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, B

Average Error: 24.8 → 0.5

Time: 8.1s

Precision: binary64

Cost: 836

Math

\[x \cdot \sqrt{y \cdot y - z \cdot z} \]

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{-288}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z \cdot -0.5}{\frac{y}{z}}\right)\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (sqrt (- (* y y) (* z z)))))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.25e-288) (* y (- x)) (* x (+ y (/ (* z -0.5) (/ y z))))))

C

double code(double x, double y, double z) {
	return x * sqrt(((y * y) - (z * z)));
}

↓

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.25e-288) {
		tmp = y * -x;
	} else {
		tmp = x * (y + ((z * -0.5) / (y / z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * sqrt(((y * y) - (z * z)))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.25d-288) then
        tmp = y * -x
    else
        tmp = x * (y + ((z * (-0.5d0)) / (y / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return x * Math.sqrt(((y * y) - (z * z)));
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.25e-288) {
		tmp = y * -x;
	} else {
		tmp = x * (y + ((z * -0.5) / (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	return x * math.sqrt(((y * y) - (z * z)))

↓

def code(x, y, z):
	tmp = 0
	if y <= 2.25e-288:
		tmp = y * -x
	else:
		tmp = x * (y + ((z * -0.5) / (y / z)))
	return tmp

Julia

function code(x, y, z)
	return Float64(x * sqrt(Float64(Float64(y * y) - Float64(z * z))))
end

↓

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.25e-288)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(x * Float64(y + Float64(Float64(z * -0.5) / Float64(y / z))));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * sqrt(((y * y) - (z * z)));
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.25e-288)
		tmp = y * -x;
	else
		tmp = x * (y + ((z * -0.5) / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(x * N[Sqrt[N[(N[(y * y), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := If[LessEqual[y, 2.25e-288], N[(y * (-x)), $MachinePrecision], N[(x * N[(y + N[(N[(z * -0.5), $MachinePrecision] / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	24.8
Target	0.6
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;y < 2.5816096488251695 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if y < 2.2500000000000001e-288

   1. Initial program 25.1

      \[x \cdot \sqrt{y \cdot y - z \cdot z} \]

   2. Taylor expanded in y around -inf 0.8

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

   3. Simplified 0.8

      \[\leadsto \color{blue}{y \cdot \left(-x\right)} \]
      Proof
      (*.f64 y (neg.f64 x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 y x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 y x))): 0 points increase in error, 0 points decrease in error

   if 2.2500000000000001e-288 < y

   1. Initial program 24.4

      \[x \cdot \sqrt{y \cdot y - z \cdot z} \]

   2. Applied egg-rr 0.5

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

   3. Applied egg-rr 0.5

      \[\leadsto x \cdot \left(\sqrt{y + z} \cdot \color{blue}{{\left({\left(y - z\right)}^{0.25}\right)}^{2}}\right) \]

   4. Taylor expanded in z around 0 2.9

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

   5. Simplified 2.9

      \[\leadsto x \cdot \color{blue}{\left(y + -0.5 \cdot \frac{z \cdot z}{y}\right)} \]
      Proof
      (+.f64 y (*.f64 -1/2 (/.f64 (*.f64 z z) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 y 0)) (*.f64 -1/2 (/.f64 (*.f64 z z) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (Rewrite<= metadata-eval (*.f64 1/2 0))) (*.f64 -1/2 (/.f64 (*.f64 z z) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (Rewrite<= div0_binary64 (/.f64 0 y)))) (*.f64 -1/2 (/.f64 (*.f64 z z) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 y)) y))) (*.f64 -1/2 (/.f64 (*.f64 z z) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 z)) y) y))) (*.f64 -1/2 (/.f64 (*.f64 z z) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) z) y) y))) (*.f64 -1/2 (/.f64 (*.f64 z z) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 -1 z) z)) y) y))) (*.f64 -1/2 (/.f64 (*.f64 z z) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 y (+.f64 (*.f64 -1 z) z))) y))) (*.f64 -1/2 (/.f64 (*.f64 z z) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 (*.f64 -1 z) y) (*.f64 z y))) y))) (*.f64 -1/2 (/.f64 (*.f64 z z) y))): 10 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (+.f64 (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 z -1)) y) (*.f64 z y)) y))) (*.f64 -1/2 (/.f64 (*.f64 z z) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 z (*.f64 -1 y))) (*.f64 z y)) y))) (*.f64 -1/2 (/.f64 (*.f64 z z) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 z (+.f64 (*.f64 -1 y) y))) y))) (*.f64 -1/2 (/.f64 (*.f64 z z) y))): 0 points increase in error, 10 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (Rewrite<= +-commutative_binary64 (+.f64 y (*.f64 -1 y)))) y))) (*.f64 -1/2 (/.f64 (*.f64 z z) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 (pow.f64 z 2) 0)) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (pow.f64 z 2))) 0) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (Rewrite<= mul0-lft_binary64 (*.f64 0 (pow.f64 z 2)))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (Rewrite<= div0_binary64 (/.f64 0 y)) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 0 1/2)) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 y)) 1/2) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) y) 1/2) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 y (*.f64 -1 y))) 1/2) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 y (*.f64 -1 y)) y) 1/2)) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (/.f64 (+.f64 y (*.f64 -1 y)) y))) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 1/2 (+.f64 y (*.f64 -1 y))) y)) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 1/2 (Rewrite=> distribute-rgt1-in_binary64 (*.f64 (+.f64 -1 1) y))) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 1/2 (*.f64 (Rewrite=> metadata-eval 0) y)) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 1/2 (Rewrite=> mul0-lft_binary64 0)) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 1/2 (Rewrite<= mul0-lft_binary64 (*.f64 0 z))) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 1/2 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) z)) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 1/2 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 -1 z) z))) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 1/2 (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 -1 1) z))) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 1/2 (*.f64 (Rewrite=> metadata-eval 0) z)) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 1/2 (Rewrite=> mul0-lft_binary64 0)) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (Rewrite=> metadata-eval 0) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (Rewrite<= mul0-rgt_binary64 (*.f64 (*.f64 1/2 (/.f64 (+.f64 y (*.f64 -1 y)) y)) 0)) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 (*.f64 1/2 (/.f64 (+.f64 y (*.f64 -1 y)) y)) (Rewrite<= metadata-eval (*.f64 1/2 0))) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 (*.f64 1/2 (/.f64 (+.f64 y (*.f64 -1 y)) y)) (*.f64 1/2 (Rewrite<= mul0-lft_binary64 (*.f64 0 y)))) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 (*.f64 1/2 (/.f64 (+.f64 y (*.f64 -1 y)) y)) (*.f64 1/2 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) y))) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (/.f64 (*.f64 (*.f64 1/2 (/.f64 (+.f64 y (*.f64 -1 y)) y)) (*.f64 1/2 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 y (*.f64 -1 y))))) y) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (*.f64 1/2 (/.f64 (+.f64 y (*.f64 -1 y)) y)) (/.f64 (*.f64 1/2 (+.f64 y (*.f64 -1 y))) y))) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (*.f64 (*.f64 1/2 (/.f64 (+.f64 y (*.f64 -1 y)) y)) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (+.f64 y (*.f64 -1 y)) y)))) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (+.f64 (*.f64 1 (pow.f64 z 2)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (*.f64 1/2 (/.f64 (+.f64 y (*.f64 -1 y)) y)) 2)) (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 y (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))) (*.f64 -1/2 (/.f64 (Rewrite=> distribute-rgt-out_binary64 (*.f64 (pow.f64 z 2) (+.f64 1 (pow.f64 (*.f64 1/2 (/.f64 (+.f64 y (*.f64 -1 y)) y)) 2)))) y))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 y (+.f64 (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y)) (*.f64 -1/2 (/.f64 (*.f64 (pow.f64 z 2) (+.f64 1 (pow.f64 (*.f64 1/2 (/.f64 (+.f64 y (*.f64 -1 y)) y)) 2))) y))))): 0 points increase in error, 0 points decrease in error
      (+.f64 y (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1/2 (/.f64 (*.f64 (pow.f64 z 2) (+.f64 1 (pow.f64 (*.f64 1/2 (/.f64 (+.f64 y (*.f64 -1 y)) y)) 2))) y)) (*.f64 1/2 (/.f64 (*.f64 z (+.f64 y (*.f64 -1 y))) y))))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 0.2

      \[\leadsto x \cdot \left(y + \color{blue}{\frac{z \cdot -0.5}{\frac{y}{z}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{-288}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{z \cdot -0.5}{\frac{y}{z}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	388

\[\begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	30.3
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2022317 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, B"
  :precision binary64

  :herbie-target
  (if (< y 2.5816096488251695e-278) (- (* x y)) (* x (* (sqrt (+ y z)) (sqrt (- y z)))))

  (* x (sqrt (- (* y y) (* z z)))))