Falkner and Boettcher, Equation (22+)

Average Error: 1.0 → 0.0

Time: 6.5s

Precision: binary64

Cost: 13888

Math

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

↓

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

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 (- 1.0 (* v v))) (- PI))
  (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 / (1.0 - (v * v))) / -((double) M_PI)) / 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 / (1.0 - (v * v))) / -Math.PI) / 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 / (1.0 - (v * v))) / -math.pi) / 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(Float64(Float64(-1.3333333333333333 / Float64(1.0 - Float64(v * v))) / Float64(-pi)) / sqrt(Float64(2.0 + Float64(Float64(v * 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 / (1.0 - (v * v))) / -pi) / 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[(N[(N[(-1.3333333333333333 / N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-Pi)), $MachinePrecision] / N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $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. Simplified 0.0

   \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]
   Proof
   (/.f64 (/.f64 4/3 (*.f64 (PI.f64) (-.f64 1 (*.f64 v v)))) (sqrt.f64 (+.f64 2 (*.f64 (*.f64 v v) -6)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 4 3)) (*.f64 (PI.f64) (-.f64 1 (*.f64 v v)))) (sqrt.f64 (+.f64 2 (*.f64 (*.f64 v v) -6)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 4 (*.f64 3 (*.f64 (PI.f64) (-.f64 1 (*.f64 v v)))))) (sqrt.f64 (+.f64 2 (*.f64 (*.f64 v v) -6)))): 3 points increase in error, 4 points decrease in error
   (/.f64 (/.f64 4 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 3 (PI.f64)) (-.f64 1 (*.f64 v v))))) (sqrt.f64 (+.f64 2 (*.f64 (*.f64 v v) -6)))): 0 points increase in error, 1 points decrease in error
   (/.f64 (/.f64 4 (*.f64 (*.f64 3 (PI.f64)) (-.f64 1 (*.f64 v v)))) (sqrt.f64 (+.f64 2 (*.f64 (*.f64 v v) (Rewrite<= metadata-eval (neg.f64 6)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 4 (*.f64 (*.f64 3 (PI.f64)) (-.f64 1 (*.f64 v v)))) (sqrt.f64 (+.f64 2 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 v v) 6)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 4 (*.f64 (*.f64 3 (PI.f64)) (-.f64 1 (*.f64 v v)))) (sqrt.f64 (+.f64 2 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 6 (*.f64 v v))))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 4 (*.f64 (*.f64 3 (PI.f64)) (-.f64 1 (*.f64 v v)))) (sqrt.f64 (Rewrite<= sub-neg_binary64 (-.f64 2 (*.f64 6 (*.f64 v v)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r*_binary64 (/.f64 4 (*.f64 (*.f64 (*.f64 3 (PI.f64)) (-.f64 1 (*.f64 v v))) (sqrt.f64 (-.f64 2 (*.f64 6 (*.f64 v v))))))): 241 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.0

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

4. Simplified 0.0

   \[\leadsto \frac{\color{blue}{\frac{\frac{-1.3333333333333333}{1 - v \cdot v}}{-\pi}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]
   Proof
   (/.f64 (/.f64 -4/3 (-.f64 1 (*.f64 v v))) (neg.f64 (PI.f64))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r*_binary64 (/.f64 -4/3 (*.f64 (-.f64 1 (*.f64 v v)) (neg.f64 (PI.f64))))): 0 points increase in error, 6 points decrease in error
   (/.f64 (Rewrite<= metadata-eval (*.f64 -4/3 1)) (*.f64 (-.f64 1 (*.f64 v v)) (neg.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 -4/3 (/.f64 1 (*.f64 (-.f64 1 (*.f64 v v)) (neg.f64 (PI.f64)))))): 246 points increase in error, 0 points decrease in error

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	13824

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

Alternative 2
Error	0.6
Cost	13568

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

Alternative 3
Error	0.6
Cost	13440

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

Alternative 4
Error	1.6
Cost	13056

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

Alternative 5
Error	0.7
Cost	13056

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

Reproduce

herbie shell --seed 2022321 
(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)))))))