Falkner and Boettcher, Appendix B, 2

Average Accuracy: 100.0% → 100.0%

Time: 6.1s

Precision: binary64

Cost: 7232

Math

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

↓

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

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(Float64(v * 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[(N[(v * v), $MachinePrecision] * -0.375), $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. Simplified 100.0%

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

   [Start] 100.0	\[ \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
   associate-*l* [=>] 100.0	\[ \color{blue}{\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)} \]
   associate-*r* [=>] 100.0	\[ \frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - \color{blue}{\left(3 \cdot v\right) \cdot v}} \cdot \left(1 - v \cdot v\right)\right) \]

3. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - \left(3 \cdot v\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right) \]
   associate-*r* [=>] 100.0	\[ \color{blue}{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \left(3 \cdot v\right) \cdot v}\right) \cdot \left(1 - v \cdot v\right)} \]
   cancel-sign-sub-inv [=>] 100.0	\[ \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \left(3 \cdot v\right) \cdot v}\right) \cdot \color{blue}{\left(1 + \left(-v\right) \cdot v\right)} \]
   distribute-lft-in [=>] 100.0	\[ \color{blue}{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \left(3 \cdot v\right) \cdot v}\right) \cdot 1 + \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \left(3 \cdot v\right) \cdot v}\right) \cdot \left(\left(-v\right) \cdot v\right)} \]

4. Simplified 100.0%

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

   [Start] 100.0	\[ \sqrt{0.125} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)} + \left(\sqrt{0.125} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(v \cdot \left(-v\right)\right) \]
   associate-*l* [=>] 100.0	\[ \sqrt{0.125} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)} + \color{blue}{\sqrt{0.125} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(v \cdot \left(-v\right)\right)\right)} \]
   distribute-lft-out [=>] 100.0	\[ \color{blue}{\sqrt{0.125} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} + \sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(v \cdot \left(-v\right)\right)\right)} \]
   +-commutative [=>] 100.0	\[ \sqrt{0.125} \cdot \color{blue}{\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(v \cdot \left(-v\right)\right) + \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)} \]
   +-commutative [<=] 100.0	\[ \sqrt{0.125} \cdot \color{blue}{\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} + \sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(v \cdot \left(-v\right)\right)\right)} \]
   *-rgt-identity [<=] 100.0	\[ \sqrt{0.125} \cdot \left(\color{blue}{\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot 1} + \sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(v \cdot \left(-v\right)\right)\right) \]
   distribute-lft-in [<=] 100.0	\[ \sqrt{0.125} \cdot \color{blue}{\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot \left(-v\right)\right)\right)} \]
   cancel-sign-sub-inv [=>] 100.0	\[ \sqrt{0.125} \cdot \left(\sqrt{\color{blue}{1 + \left(-3\right) \cdot \left(v \cdot v\right)}} \cdot \left(1 + v \cdot \left(-v\right)\right)\right) \]
   metadata-eval [=>] 100.0	\[ \sqrt{0.125} \cdot \left(\sqrt{1 + \color{blue}{-3} \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot \left(-v\right)\right)\right) \]
   *-commutative [<=] 100.0	\[ \sqrt{0.125} \cdot \left(\sqrt{1 + \color{blue}{\left(v \cdot v\right) \cdot -3}} \cdot \left(1 + v \cdot \left(-v\right)\right)\right) \]
   associate-*r* [<=] 100.0	\[ \sqrt{0.125} \cdot \left(\sqrt{1 + \color{blue}{v \cdot \left(v \cdot -3\right)}} \cdot \left(1 + v \cdot \left(-v\right)\right)\right) \]
   *-commutative [=>] 100.0	\[ \sqrt{0.125} \cdot \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \left(1 + \color{blue}{\left(-v\right) \cdot v}\right)\right) \]

5. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \sqrt{0.125} \cdot \left(\sqrt{1 + v \cdot \left(v \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right) \]
   associate-*r* [=>] 100.0	\[ \color{blue}{\left(\sqrt{0.125} \cdot \sqrt{1 + v \cdot \left(v \cdot -3\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   cancel-sign-sub-inv [=>] 100.0	\[ \left(\sqrt{0.125} \cdot \sqrt{1 + v \cdot \left(v \cdot -3\right)}\right) \cdot \color{blue}{\left(1 + \left(-v\right) \cdot v\right)} \]
   distribute-rgt-in [=>] 100.0	\[ \color{blue}{1 \cdot \left(\sqrt{0.125} \cdot \sqrt{1 + v \cdot \left(v \cdot -3\right)}\right) + \left(\left(-v\right) \cdot v\right) \cdot \left(\sqrt{0.125} \cdot \sqrt{1 + v \cdot \left(v \cdot -3\right)}\right)} \]
   *-un-lft-identity [<=] 100.0	\[ \color{blue}{\sqrt{0.125} \cdot \sqrt{1 + v \cdot \left(v \cdot -3\right)}} + \left(\left(-v\right) \cdot v\right) \cdot \left(\sqrt{0.125} \cdot \sqrt{1 + v \cdot \left(v \cdot -3\right)}\right) \]
   sqrt-unprod [=>] 100.0	\[ \color{blue}{\sqrt{0.125 \cdot \left(1 + v \cdot \left(v \cdot -3\right)\right)}} + \left(\left(-v\right) \cdot v\right) \cdot \left(\sqrt{0.125} \cdot \sqrt{1 + v \cdot \left(v \cdot -3\right)}\right) \]
   +-commutative [=>] 100.0	\[ \sqrt{0.125 \cdot \color{blue}{\left(v \cdot \left(v \cdot -3\right) + 1\right)}} + \left(\left(-v\right) \cdot v\right) \cdot \left(\sqrt{0.125} \cdot \sqrt{1 + v \cdot \left(v \cdot -3\right)}\right) \]
   associate-*r* [=>] 100.0	\[ \sqrt{0.125 \cdot \left(\color{blue}{\left(v \cdot v\right) \cdot -3} + 1\right)} + \left(\left(-v\right) \cdot v\right) \cdot \left(\sqrt{0.125} \cdot \sqrt{1 + v \cdot \left(v \cdot -3\right)}\right) \]
   fma-def [=>] 100.0	\[ \sqrt{0.125 \cdot \color{blue}{\mathsf{fma}\left(v \cdot v, -3, 1\right)}} + \left(\left(-v\right) \cdot v\right) \cdot \left(\sqrt{0.125} \cdot \sqrt{1 + v \cdot \left(v \cdot -3\right)}\right) \]
   distribute-lft-neg-in [<=] 100.0	\[ \sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)} + \color{blue}{\left(-v \cdot v\right)} \cdot \left(\sqrt{0.125} \cdot \sqrt{1 + v \cdot \left(v \cdot -3\right)}\right) \]
   sqrt-unprod [=>] 100.0	\[ \sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)} + \left(-v \cdot v\right) \cdot \color{blue}{\sqrt{0.125 \cdot \left(1 + v \cdot \left(v \cdot -3\right)\right)}} \]
   +-commutative [=>] 100.0	\[ \sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)} + \left(-v \cdot v\right) \cdot \sqrt{0.125 \cdot \color{blue}{\left(v \cdot \left(v \cdot -3\right) + 1\right)}} \]
   associate-*r* [=>] 100.0	\[ \sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)} + \left(-v \cdot v\right) \cdot \sqrt{0.125 \cdot \left(\color{blue}{\left(v \cdot v\right) \cdot -3} + 1\right)} \]
   fma-def [=>] 100.0	\[ \sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)} + \left(-v \cdot v\right) \cdot \sqrt{0.125 \cdot \color{blue}{\mathsf{fma}\left(v \cdot v, -3, 1\right)}} \]

6. Simplified 100.0%

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

   [Start] 100.0	\[ \sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)} + \left(-v \cdot v\right) \cdot \sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)} \]
   distribute-rgt1-in [=>] 100.0	\[ \color{blue}{\left(\left(-v \cdot v\right) + 1\right) \cdot \sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}} \]
   +-commutative [<=] 100.0	\[ \color{blue}{\left(1 + \left(-v \cdot v\right)\right)} \cdot \sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)} \]
   sub-neg [<=] 100.0	\[ \color{blue}{\left(1 - v \cdot v\right)} \cdot \sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)} \]
   *-commutative [=>] 100.0	\[ \color{blue}{\sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)} \cdot \left(1 - v \cdot v\right)} \]

7. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}\right)} - 1\right)} \cdot \left(1 - v \cdot v\right) \]
   Proof

   [Start] 100.0	\[ \sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)} \cdot \left(1 - v \cdot v\right) \]
   expm1-log1p-u [=>] 100.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}\right)\right)} \cdot \left(1 - v \cdot v\right) \]
   expm1-udef [=>] 98.4	\[ \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}\right)} - 1\right)} \cdot \left(1 - v \cdot v\right) \]

8. Simplified 100.0%

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

   [Start] 98.4	\[ \left(e^{\mathsf{log1p}\left(\sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}\right)} - 1\right) \cdot \left(1 - v \cdot v\right) \]
   expm1-def [=>] 100.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}\right)\right)} \cdot \left(1 - v \cdot v\right) \]
   expm1-log1p [=>] 100.0	\[ \color{blue}{\sqrt{0.125 \cdot \mathsf{fma}\left(v \cdot v, -3, 1\right)}} \cdot \left(1 - v \cdot v\right) \]
   *-commutative [=>] 100.0	\[ \sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot 0.125}} \cdot \left(1 - v \cdot v\right) \]
   unpow2 [<=] 100.0	\[ \sqrt{\mathsf{fma}\left(\color{blue}{{v}^{2}}, -3, 1\right) \cdot 0.125} \cdot \left(1 - v \cdot v\right) \]
   fma-udef [=>] 100.0	\[ \sqrt{\color{blue}{\left({v}^{2} \cdot -3 + 1\right)} \cdot 0.125} \cdot \left(1 - v \cdot v\right) \]
   distribute-rgt1-in [<=] 100.0	\[ \sqrt{\color{blue}{0.125 + \left({v}^{2} \cdot -3\right) \cdot 0.125}} \cdot \left(1 - v \cdot v\right) \]
   rem-square-sqrt [<=] 0.0	\[ \sqrt{0.125 + \left({v}^{2} \cdot \color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)}\right) \cdot 0.125} \cdot \left(1 - v \cdot v\right) \]
   associate-*l* [=>] 0.0	\[ \sqrt{0.125 + \color{blue}{{v}^{2} \cdot \left(\left(\sqrt{-3} \cdot \sqrt{-3}\right) \cdot 0.125\right)}} \cdot \left(1 - v \cdot v\right) \]
   unpow2 [=>] 0.0	\[ \sqrt{0.125 + \color{blue}{\left(v \cdot v\right)} \cdot \left(\left(\sqrt{-3} \cdot \sqrt{-3}\right) \cdot 0.125\right)} \cdot \left(1 - v \cdot v\right) \]
   rem-square-sqrt [=>] 100.0	\[ \sqrt{0.125 + \left(v \cdot v\right) \cdot \left(\color{blue}{-3} \cdot 0.125\right)} \cdot \left(1 - v \cdot v\right) \]
   metadata-eval [=>] 100.0	\[ \sqrt{0.125 + \left(v \cdot v\right) \cdot \color{blue}{-0.375}} \cdot \left(1 - v \cdot v\right) \]

9. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	6976

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

Alternative 2
Accuracy	99.0%
Cost	6848

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

Alternative 3
Accuracy	98.9%
Cost	6464

\[\sqrt{0.125} \]

Reproduce

herbie shell --seed 2023135 
(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))))