Falkner and Boettcher, Appendix B, 2

Average Accuracy: 100.0% → 100.0%

Time: 3.5s

Precision: binary64

Cost: 7232

Math

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

↓

\[\sqrt{0.125 + v \cdot \left(v \cdot -0.375\right)} \cdot \left(1 - v \cdot v\right) \]

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 (+ 0.125 (* v (* v -0.375)))) (- 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((0.125 + (v * (v * -0.375)))) * (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((0.125d0 + (v * (v * (-0.375d0))))) * (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((0.125 + (v * (v * -0.375)))) * (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((0.125 + (v * (v * -0.375)))) * (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(0.125 + Float64(v * Float64(v * -0.375)))) * 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((0.125 + (v * (v * -0.375)))) * (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[(0.125 + N[(v * N[(v * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(0 + \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot 0.125}\right)} \cdot \left(1 - v \cdot v\right) \]
   Step-by-step derivation

   [Start] 100.0	\[ \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
   add-log-exp [=>] 100.0	\[ \color{blue}{\log \left(e^{\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}\right)} \cdot \left(1 - v \cdot v\right) \]
   *-un-lft-identity [=>] 100.0	\[ \log \color{blue}{\left(1 \cdot e^{\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}\right)} \cdot \left(1 - v \cdot v\right) \]
   log-prod [=>] 100.0	\[ \color{blue}{\left(\log 1 + \log \left(e^{\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}\right)\right)} \cdot \left(1 - v \cdot v\right) \]
   metadata-eval [=>] 100.0	\[ \left(\color{blue}{0} + \log \left(e^{\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}\right)\right) \cdot \left(1 - v \cdot v\right) \]
   add-log-exp [<=] 100.0	\[ \left(0 + \color{blue}{\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}\right) \cdot \left(1 - v \cdot v\right) \]
   add-sqr-sqrt [=>] 98.4	\[ \left(0 + \color{blue}{\sqrt{\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}} \cdot \sqrt{\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}\right) \cdot \left(1 - v \cdot v\right) \]
   sqrt-unprod [=>] 100.0	\[ \left(0 + \color{blue}{\sqrt{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}}\right) \cdot \left(1 - v \cdot v\right) \]
   *-commutative [=>] 100.0	\[ \left(0 + \sqrt{\color{blue}{\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{2}}{4}\right)} \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}\right) \cdot \left(1 - v \cdot v\right) \]
   *-commutative [=>] 100.0	\[ \left(0 + \sqrt{\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{2}}{4}\right) \cdot \color{blue}{\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{2}}{4}\right)}}\right) \cdot \left(1 - v \cdot v\right) \]
   swap-sqr [=>] 100.0	\[ \left(0 + \sqrt{\color{blue}{\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{4}\right)}}\right) \cdot \left(1 - v \cdot v\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{0.125 + v \cdot \left(v \cdot -0.375\right)}} \cdot \left(1 - v \cdot v\right) \]
   Step-by-step derivation

   [Start] 100.0	\[ \left(0 + \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot 0.125}\right) \cdot \left(1 - v \cdot v\right) \]
   +-lft-identity [=>] 100.0	\[ \color{blue}{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot 0.125}} \cdot \left(1 - v \cdot v\right) \]
   fma-udef [=>] 100.0	\[ \sqrt{\color{blue}{\left(\left(v \cdot v\right) \cdot -3 + 1\right)} \cdot 0.125} \cdot \left(1 - v \cdot v\right) \]
   distribute-lft1-in [<=] 100.0	\[ \sqrt{\color{blue}{\left(\left(v \cdot v\right) \cdot -3\right) \cdot 0.125 + 0.125}} \cdot \left(1 - v \cdot v\right) \]
   +-commutative [=>] 100.0	\[ \sqrt{\color{blue}{0.125 + \left(\left(v \cdot v\right) \cdot -3\right) \cdot 0.125}} \cdot \left(1 - v \cdot v\right) \]
   associate-*l* [=>] 100.0	\[ \sqrt{0.125 + \color{blue}{\left(v \cdot v\right) \cdot \left(-3 \cdot 0.125\right)}} \cdot \left(1 - v \cdot v\right) \]
   metadata-eval [=>] 100.0	\[ \sqrt{0.125 + \left(v \cdot v\right) \cdot \color{blue}{-0.375}} \cdot \left(1 - v \cdot v\right) \]
   associate-*l* [=>] 100.0	\[ \sqrt{0.125 + \color{blue}{v \cdot \left(v \cdot -0.375\right)}} \cdot \left(1 - v \cdot v\right) \]

4. Final simplification 100.0%

   \[\leadsto \sqrt{0.125 + v \cdot \left(v \cdot -0.375\right)} \cdot \left(1 - v \cdot v\right) \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	6976

\[\left(1 - v \cdot v\right) \cdot \left(0.25 \cdot \sqrt{2}\right) \]

Alternative 2
Accuracy	99.1%
Cost	6592

\[0.25 \cdot \sqrt{2} \]

Reproduce

herbie shell --seed 2023157 
(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))))