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

Average Error: 24.3 → 0.5

Time: 8.5s

Precision: binary64

Cost: 13508

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 9.6 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \left(\frac{z}{\frac{y}{z \cdot 0.5}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.6e-275)
   (* x (- (/ z (/ y (* z 0.5))) y))
   (* x (* (sqrt (+ y z)) (sqrt (- 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 <= 9.6e-275) {
		tmp = x * ((z / (y / (z * 0.5))) - y);
	} else {
		tmp = x * (sqrt((y + z)) * sqrt((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 <= 9.6d-275) then
        tmp = x * ((z / (y / (z * 0.5d0))) - y)
    else
        tmp = x * (sqrt((y + z)) * sqrt((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 <= 9.6e-275) {
		tmp = x * ((z / (y / (z * 0.5))) - y);
	} else {
		tmp = x * (Math.sqrt((y + z)) * Math.sqrt((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 <= 9.6e-275:
		tmp = x * ((z / (y / (z * 0.5))) - y)
	else:
		tmp = x * (math.sqrt((y + z)) * math.sqrt((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 <= 9.6e-275)
		tmp = Float64(x * Float64(Float64(z / Float64(y / Float64(z * 0.5))) - y));
	else
		tmp = Float64(x * Float64(sqrt(Float64(y + z)) * sqrt(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 <= 9.6e-275)
		tmp = x * ((z / (y / (z * 0.5))) - y);
	else
		tmp = x * (sqrt((y + z)) * sqrt((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, 9.6e-275], N[(x * N[(N[(z / N[(y / N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Sqrt[N[(y + z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(y - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	24.3
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 < 9.59999999999999962e-275

   1. Initial program 24.4

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

   2. Taylor expanded in y around -inf 3.4

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

   3. Simplified 3.4

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

   4. Taylor expanded in y around 0 3.4

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

   5. Simplified 0.5

      \[\leadsto x \cdot \left(\color{blue}{\frac{z}{\frac{y}{0.5 \cdot z}}} - y\right) \]
      Proof
      (/.f64 z (/.f64 y (*.f64 1/2 z))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 z (*.f64 1/2 z)) y)): 38 points increase in error, 41 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 1/2 z) z)) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 z z))) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 1/2 (Rewrite<= unpow2_binary64 (pow.f64 z 2))) y): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (pow.f64 z 2) y))): 0 points increase in error, 1 points decrease in error

   if 9.59999999999999962e-275 < y

   1. Initial program 24.1

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

   2. Applied egg-rr 0.4

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.6 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \left(\frac{z}{\frac{y}{z \cdot 0.5}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	964

\[\begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-293}:\\ \;\;\;\;x \cdot \left(\frac{z}{\frac{y}{z \cdot 0.5}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(z \cdot \frac{z}{y}\right) \cdot -0.5\right) + y \cdot x\\ \end{array} \]

Alternative 2
Error	0.5
Cost	836

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-293}:\\ \;\;\;\;x \cdot \left(\frac{z}{\frac{y}{z \cdot 0.5}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Error	1.0
Cost	388

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-235}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4
Error	30.2
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2022316 
(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)))))