Falkner and Boettcher, Appendix B, 2

Average Error: 0.02% → 0.02%

Time: 9.5s

Precision: binary64

Cost: 7360

Math

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

↓

\[\frac{1 - v \cdot v}{\frac{4}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]

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
 (/ (- 1.0 (* v v)) (/ 4.0 (sqrt (+ 2.0 (* (* v v) -6.0))))))

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 (1.0 - (v * v)) / (4.0 / sqrt((2.0 + ((v * v) * -6.0))));
}

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 = (1.0d0 - (v * v)) / (4.0d0 / sqrt((2.0d0 + ((v * v) * (-6.0d0)))))
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 (1.0 - (v * v)) / (4.0 / Math.sqrt((2.0 + ((v * v) * -6.0))));
}

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 (1.0 - (v * v)) / (4.0 / math.sqrt((2.0 + ((v * v) * -6.0))))

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(Float64(1.0 - Float64(v * v)) / Float64(4.0 / sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0)))))
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 = (1.0 - (v * v)) / (4.0 / sqrt((2.0 + ((v * v) * -6.0))));
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[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] / N[(4.0 / N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.02

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

2. Applied egg-rr 0.02

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

3. Applied egg-rr 0.03

   \[\leadsto \frac{1 - v \cdot v}{\color{blue}{e^{\mathsf{log1p}\left(\frac{4}{\sqrt{2 + -6 \cdot \left(v \cdot v\right)}}\right)} - 1}} \]

4. Simplified 0.02

   \[\leadsto \frac{1 - v \cdot v}{\color{blue}{\frac{4}{\sqrt{2 + -6 \cdot \left(v \cdot v\right)}}}} \]
   Proof

   [Start] 0.03	\[ \frac{1 - v \cdot v}{e^{\mathsf{log1p}\left(\frac{4}{\sqrt{2 + -6 \cdot \left(v \cdot v\right)}}\right)} - 1} \]
   expm1-def [=>] 0.03	\[ \frac{1 - v \cdot v}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{4}{\sqrt{2 + -6 \cdot \left(v \cdot v\right)}}\right)\right)}} \]
   expm1-log1p [=>] 0.02	\[ \frac{1 - v \cdot v}{\color{blue}{\frac{4}{\sqrt{2 + -6 \cdot \left(v \cdot v\right)}}}} \]

5. Final simplification 0.02

   \[\leadsto \frac{1 - v \cdot v}{\frac{4}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]

Alternatives

Alternative 1
Error	0.01%
Cost	7360

\[\left(1 - v \cdot v\right) \cdot \left(\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot 0.25\right) \]

Alternative 2
Error	0.43%
Cost	7232

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

Alternative 3
Error	0.42%
Cost	7104

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

Alternative 4
Error	0.42%
Cost	6976

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

Alternative 5
Error	0.99%
Cost	6848

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

Alternative 6
Error	1.02%
Cost	6464

\[\sqrt{0.125} \]

Reproduce

herbie shell --seed 2023090 
(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))))