Falkner and Boettcher, Appendix B, 2

Average Accuracy: 100.0% → 99.5%

Time: 5.9s

Precision: binary64

Cost: 6848

Math

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

↓

\[\sqrt{0.125 + \left(v \cdot v\right) \cdot -0.625} \]

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) -0.625))))

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) * -0.625)));
}

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) * (-0.625d0))))
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) * -0.625)));
}

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) * -0.625)))

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 sqrt(Float64(0.125 + Float64(Float64(v * v) * -0.625)))
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) * -0.625)));
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[Sqrt[N[(0.125 + N[(N[(v * v), $MachinePrecision] * -0.625), $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%

   \[\leadsto \color{blue}{\sqrt{\left(\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot {\left(1 - v \cdot v\right)}^{2}\right) \cdot 0.125}} \]
   Proof

   [Start] 100.0	\[ \frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - \left(3 \cdot v\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right) \]
   add-sqr-sqrt [=>] 98.5	\[ \color{blue}{\sqrt{\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - \left(3 \cdot v\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right)} \cdot \sqrt{\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - \left(3 \cdot v\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right)}} \]
   sqrt-unprod [=>] 100.0	\[ \color{blue}{\sqrt{\left(\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - \left(3 \cdot v\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - \left(3 \cdot v\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right)\right)}} \]
   *-commutative [=>] 100.0	\[ \sqrt{\color{blue}{\left(\left(\sqrt{1 - \left(3 \cdot v\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right) \cdot \frac{\sqrt{2}}{4}\right)} \cdot \left(\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - \left(3 \cdot v\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   *-commutative [=>] 100.0	\[ \sqrt{\left(\left(\sqrt{1 - \left(3 \cdot v\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right) \cdot \frac{\sqrt{2}}{4}\right) \cdot \color{blue}{\left(\left(\sqrt{1 - \left(3 \cdot v\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right) \cdot \frac{\sqrt{2}}{4}\right)}} \]
   swap-sqr [=>] 100.0	\[ \sqrt{\color{blue}{\left(\left(\sqrt{1 - \left(3 \cdot v\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(3 \cdot v\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{4}\right)}} \]

4. Taylor expanded in v around 0 99.5%

   \[\leadsto \sqrt{\color{blue}{\left(1 + -5 \cdot {v}^{2}\right)} \cdot 0.125} \]

5. Simplified 99.5%

   \[\leadsto \sqrt{\color{blue}{\left(1 + -5 \cdot \left(v \cdot v\right)\right)} \cdot 0.125} \]
   Proof

   [Start] 99.5	\[ \sqrt{\left(1 + -5 \cdot {v}^{2}\right) \cdot 0.125} \]
   unpow2 [=>] 99.5	\[ \sqrt{\left(1 + -5 \cdot \color{blue}{\left(v \cdot v\right)}\right) \cdot 0.125} \]

6. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(-5, v \cdot v, 1\right) \cdot 0.125}\right)} - 1} \]
   Proof

   [Start] 99.5	\[ \sqrt{\left(1 + -5 \cdot \left(v \cdot v\right)\right) \cdot 0.125} \]
   expm1-log1p-u [=>] 99.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(1 + -5 \cdot \left(v \cdot v\right)\right) \cdot 0.125}\right)\right)} \]
   expm1-udef [=>] 98.0	\[ \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\left(1 + -5 \cdot \left(v \cdot v\right)\right) \cdot 0.125}\right)} - 1} \]
   +-commutative [=>] 98.0	\[ e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(-5 \cdot \left(v \cdot v\right) + 1\right)} \cdot 0.125}\right)} - 1 \]
   fma-def [=>] 98.0	\[ e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\mathsf{fma}\left(-5, v \cdot v, 1\right)} \cdot 0.125}\right)} - 1 \]

7. Simplified 99.5%

   \[\leadsto \color{blue}{\sqrt{0.125 + \left(v \cdot v\right) \cdot -0.625}} \]
   Proof

   [Start] 98.0	\[ e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(-5, v \cdot v, 1\right) \cdot 0.125}\right)} - 1 \]
   expm1-def [=>] 99.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(-5, v \cdot v, 1\right) \cdot 0.125}\right)\right)} \]
   expm1-log1p [=>] 99.5	\[ \color{blue}{\sqrt{\mathsf{fma}\left(-5, v \cdot v, 1\right) \cdot 0.125}} \]
   unpow2 [<=] 99.5	\[ \sqrt{\mathsf{fma}\left(-5, \color{blue}{{v}^{2}}, 1\right) \cdot 0.125} \]
   fma-udef [=>] 99.5	\[ \sqrt{\color{blue}{\left(-5 \cdot {v}^{2} + 1\right)} \cdot 0.125} \]
   unpow2 [=>] 99.5	\[ \sqrt{\left(-5 \cdot \color{blue}{\left(v \cdot v\right)} + 1\right) \cdot 0.125} \]
   distribute-lft1-in [<=] 99.5	\[ \sqrt{\color{blue}{\left(-5 \cdot \left(v \cdot v\right)\right) \cdot 0.125 + 0.125}} \]
   +-commutative [=>] 99.5	\[ \sqrt{\color{blue}{0.125 + \left(-5 \cdot \left(v \cdot v\right)\right) \cdot 0.125}} \]
   *-commutative [=>] 99.5	\[ \sqrt{0.125 + \color{blue}{\left(\left(v \cdot v\right) \cdot -5\right)} \cdot 0.125} \]
   unpow2 [<=] 99.5	\[ \sqrt{0.125 + \left(\color{blue}{{v}^{2}} \cdot -5\right) \cdot 0.125} \]
   associate-*l* [=>] 99.5	\[ \sqrt{0.125 + \color{blue}{{v}^{2} \cdot \left(-5 \cdot 0.125\right)}} \]
   unpow2 [=>] 99.5	\[ \sqrt{0.125 + \color{blue}{\left(v \cdot v\right)} \cdot \left(-5 \cdot 0.125\right)} \]
   metadata-eval [=>] 99.5	\[ \sqrt{0.125 + \left(v \cdot v\right) \cdot \color{blue}{-0.625}} \]

8. Final simplification 99.5%

   \[\leadsto \sqrt{0.125 + \left(v \cdot v\right) \cdot -0.625} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	6464

\[\sqrt{0.125} \]

Reproduce

herbie shell --seed 2023140 
(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))))