Falkner and Boettcher, Appendix B, 2

Average Error: 0.0 → 0.0

Time: 6.5s

Precision: binary64

Cost: 13760

Math

\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

↓

\[\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \left(\sqrt{0.125} \cdot \left(1 - v \cdot v\right)\right) \]

FPCore

(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))

↓

(FPCore (v)
 :precision binary64
 (* (sqrt (+ 1.0 (* (* v v) -3.0))) (* (sqrt 0.125) (- 1.0 (* v v)))))

C

double code(double v) {
	return ((sqrt(2.0) / 4.0) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}

↓

double code(double v) {
	return sqrt((1.0 + ((v * v) * -3.0))) * (sqrt(0.125) * (1.0 - (v * v)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = ((sqrt(2.0d0) / 4.0d0) * sqrt((1.0d0 - (3.0d0 * (v * v))))) * (1.0d0 - (v * v))
end function

↓

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt((1.0d0 + ((v * v) * (-3.0d0)))) * (sqrt(0.125d0) * (1.0d0 - (v * v)))
end function

Java

public static double code(double v) {
	return ((Math.sqrt(2.0) / 4.0) * Math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}

↓

public static double code(double v) {
	return Math.sqrt((1.0 + ((v * v) * -3.0))) * (Math.sqrt(0.125) * (1.0 - (v * v)));
}

Python

def code(v):
	return ((math.sqrt(2.0) / 4.0) * math.sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v))

↓

def code(v):
	return math.sqrt((1.0 + ((v * v) * -3.0))) * (math.sqrt(0.125) * (1.0 - (v * v)))

Julia

function code(v)
	return Float64(Float64(Float64(sqrt(2.0) / 4.0) * sqrt(Float64(1.0 - Float64(3.0 * Float64(v * v))))) * Float64(1.0 - Float64(v * v)))
end

↓

function code(v)
	return Float64(sqrt(Float64(1.0 + Float64(Float64(v * v) * -3.0))) * Float64(sqrt(0.125) * Float64(1.0 - Float64(v * v))))
end

MATLAB

function tmp = code(v)
	tmp = ((sqrt(2.0) / 4.0) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
end

↓

function tmp = code(v)
	tmp = sqrt((1.0 + ((v * v) * -3.0))) * (sqrt(0.125) * (1.0 - (v * v)));
end

Wolfram

code[v_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[v_] := N[(N[Sqrt[N[(1.0 + N[(N[(v * v), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[0.125], $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

   \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

2. Simplified 0.0

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - \left(3 \cdot v\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right)} \]
   Proof

   [Start] 0.0	\[ \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
   associate-*l* [=>] 0.0	\[ \color{blue}{\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)} \]
   associate-*r* [=>] 0.0	\[ \frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - \color{blue}{\left(3 \cdot v\right) \cdot v}} \cdot \left(1 - v \cdot v\right)\right) \]

3. Applied egg-rr 0.0

   \[\leadsto \color{blue}{\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125} + \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right) \cdot \left(v \cdot \left(-v\right)\right)} \]

4. Simplified 0.0

   \[\leadsto \color{blue}{\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \left(\sqrt{0.125} \cdot \left(1 - v \cdot v\right)\right)} \]
   Proof

   [Start] 0.0	\[ \sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125} + \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right) \cdot \left(v \cdot \left(-v\right)\right) \]
   *-commutative [=>] 0.0	\[ \sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125} + \color{blue}{\left(v \cdot \left(-v\right)\right) \cdot \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right)} \]
   *-lft-identity [<=] 0.0	\[ \color{blue}{\left(1 \cdot \sqrt{1 + v \cdot \left(v \cdot -3\right)}\right)} \cdot \sqrt{0.125} + \left(v \cdot \left(-v\right)\right) \cdot \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right) \]
   associate-*l* [=>] 0.0	\[ \color{blue}{1 \cdot \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right)} + \left(v \cdot \left(-v\right)\right) \cdot \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right) \]
   distribute-rgt-in [<=] 0.0	\[ \color{blue}{\left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right) \cdot \left(1 + v \cdot \left(-v\right)\right)} \]
   *-commutative [=>] 0.0	\[ \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right) \cdot \left(1 + \color{blue}{\left(-v\right) \cdot v}\right) \]
   cancel-sign-sub-inv [<=] 0.0	\[ \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \sqrt{0.125}\right) \cdot \color{blue}{\left(1 - v \cdot v\right)} \]
   associate-*r* [<=] 0.0	\[ \color{blue}{\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \left(\sqrt{0.125} \cdot \left(1 - v \cdot v\right)\right)} \]
   associate-*r* [=>] 0.0	\[ \sqrt{1 + \color{blue}{\left(v \cdot v\right) \cdot -3}} \cdot \left(\sqrt{0.125} \cdot \left(1 - v \cdot v\right)\right) \]

5. Final simplification 0.0

   \[\leadsto \sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \left(\sqrt{0.125} \cdot \left(1 - v \cdot v\right)\right) \]

Alternatives

Alternative 1
Error	0.0
Cost	13632

\[\left(1 - v \cdot v\right) \cdot \sqrt{0.125 \cdot \mathsf{fma}\left(v, v \cdot -3, 1\right)} \]

Alternative 2
Error	0.3
Cost	7104

\[\frac{\sqrt{2}}{\frac{4}{1 + \left(v \cdot v\right) \cdot -2.5}} \]

Alternative 3
Error	0.3
Cost	6976

\[\sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot -0.625\right) \]

Alternative 4
Error	0.6
Cost	6464

\[\sqrt{0.125} \]

Reproduce

herbie shell --seed 2023066 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))