Falkner and Boettcher, Equation (22+)

Average Error: 1.0 → 0.0

Time: 6.1s

Precision: binary64

Cost: 13824

Math

\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

↓

\[\frac{1.3333333333333333}{\left(\pi \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 + v \cdot \left(v \cdot -6\right)}} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))

↓

(FPCore (v)
 :precision binary64
 (/
  1.3333333333333333
  (* (* PI (- 1.0 (* v v))) (sqrt (+ 2.0 (* v (* v -6.0)))))))

C

double code(double v) {
	return 4.0 / (((3.0 * ((double) M_PI)) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
}

↓

double code(double v) {
	return 1.3333333333333333 / ((((double) M_PI) * (1.0 - (v * v))) * sqrt((2.0 + (v * (v * -6.0)))));
}

Java

public static double code(double v) {
	return 4.0 / (((3.0 * Math.PI) * (1.0 - (v * v))) * Math.sqrt((2.0 - (6.0 * (v * v)))));
}

↓

public static double code(double v) {
	return 1.3333333333333333 / ((Math.PI * (1.0 - (v * v))) * Math.sqrt((2.0 + (v * (v * -6.0)))));
}

Python

def code(v):
	return 4.0 / (((3.0 * math.pi) * (1.0 - (v * v))) * math.sqrt((2.0 - (6.0 * (v * v)))))

↓

def code(v):
	return 1.3333333333333333 / ((math.pi * (1.0 - (v * v))) * math.sqrt((2.0 + (v * (v * -6.0)))))

Julia

function code(v)
	return Float64(4.0 / Float64(Float64(Float64(3.0 * pi) * Float64(1.0 - Float64(v * v))) * sqrt(Float64(2.0 - Float64(6.0 * Float64(v * v))))))
end

↓

function code(v)
	return Float64(1.3333333333333333 / Float64(Float64(pi * Float64(1.0 - Float64(v * v))) * sqrt(Float64(2.0 + Float64(v * Float64(v * -6.0))))))
end

MATLAB

function tmp = code(v)
	tmp = 4.0 / (((3.0 * pi) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
end

↓

function tmp = code(v)
	tmp = 1.3333333333333333 / ((pi * (1.0 - (v * v))) * sqrt((2.0 + (v * (v * -6.0)))));
end

Wolfram

code[v_] := N[(4.0 / N[(N[(N[(3.0 * Pi), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 - N[(6.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[v_] := N[(1.3333333333333333 / N[(N[(Pi * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 + N[(v * N[(v * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 1.0

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

2. Applied egg-rr 1.0

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}\right)} - 1} \]

3. Simplified 0.0

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\left(\pi \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 + v \cdot \left(v \cdot -6\right)}}} \]
   Proof

   [Start] 1.0	\[ e^{\mathsf{log1p}\left(\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}\right)} - 1 \]
   expm1-def [=>] 0.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}\right)\right)} \]
   expm1-log1p [=>] 0.0	\[ \color{blue}{\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]
   associate-/r* [<=] 0.0	\[ \color{blue}{\frac{1.3333333333333333}{\left(\pi \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]
   associate-*l* [=>] 0.0	\[ \frac{1.3333333333333333}{\left(\pi \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 + \color{blue}{v \cdot \left(v \cdot -6\right)}}} \]

4. Final simplification 0.0

   \[\leadsto \frac{1.3333333333333333}{\left(\pi \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 + v \cdot \left(v \cdot -6\right)}} \]

Alternatives

Alternative 1
Error	0.7
Cost	13568

\[\frac{1.3333333333333333 \cdot \frac{1}{\pi}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

Alternative 2
Error	0.7
Cost	13440

\[\frac{1.3333333333333333}{\pi \cdot \sqrt{2 + v \cdot \left(v \cdot -6\right)}} \]

Alternative 3
Error	0.7
Cost	13056

\[\frac{1.3333333333333333}{\frac{\pi}{\sqrt{0.5}}} \]

Alternative 4
Error	1.7
Cost	12928

\[\frac{\sqrt{0.8888888888888888}}{\pi} \]

Reproduce

herbie shell --seed 2023053 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))