Falkner and Boettcher, Equation (22+)

Average Error: 1.0 → 0.0

Time: 6.3s

Precision: binary64

Cost: 26496

Math

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

↓

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

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) (sqrt (fma v (* v -6.0) 2.0)))
  (/ 1.0 (fma v v -1.0))))

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)) / sqrt(fma(v, (v * -6.0), 2.0))) * (1.0 / fma(v, v, -1.0));
}

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 / pi) / sqrt(fma(v, Float64(v * -6.0), 2.0))) * Float64(1.0 / fma(v, v, -1.0)))
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 / Pi), $MachinePrecision] / N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(v * v + -1.0), $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

   \[\leadsto \color{blue}{\frac{-1.3333333333333333}{\left(\pi \cdot \sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}\right) \cdot \mathsf{fma}\left(v, v, -1\right)}} \]
   Proof
   (/.f64 -4/3 (*.f64 (*.f64 (PI.f64) (sqrt.f64 (fma.f64 v (*.f64 v -6) 2))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= metadata-eval (/.f64 4/3 -1)) (*.f64 (*.f64 (PI.f64) (sqrt.f64 (fma.f64 v (*.f64 v -6) 2))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 4 3)) -1) (*.f64 (*.f64 (PI.f64) (sqrt.f64 (fma.f64 v (*.f64 v -6) 2))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (/.f64 4 3) -1) (*.f64 (*.f64 (PI.f64) (sqrt.f64 (fma.f64 v (*.f64 v (Rewrite<= metadata-eval (neg.f64 6))) 2))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (/.f64 4 3) -1) (*.f64 (*.f64 (PI.f64) (sqrt.f64 (fma.f64 v (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 v 6))) 2))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (/.f64 4 3) -1) (*.f64 (*.f64 (PI.f64) (sqrt.f64 (fma.f64 v (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 6 v))) 2))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (/.f64 4 3) -1) (*.f64 (*.f64 (PI.f64) (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 v (neg.f64 (*.f64 6 v))) 2)))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (/.f64 4 3) -1) (*.f64 (*.f64 (PI.f64) (sqrt.f64 (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 v (*.f64 6 v)))) 2))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (/.f64 4 3) -1) (*.f64 (*.f64 (PI.f64) (sqrt.f64 (+.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 6 v) v))) 2))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (/.f64 4 3) -1) (*.f64 (*.f64 (PI.f64) (sqrt.f64 (+.f64 (neg.f64 (Rewrite<= associate-*r*_binary64 (*.f64 6 (*.f64 v v)))) 2))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (/.f64 4 3) -1) (*.f64 (*.f64 (PI.f64) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 2 (neg.f64 (*.f64 6 (*.f64 v v))))))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (/.f64 4 3) -1) (*.f64 (*.f64 (PI.f64) (sqrt.f64 (Rewrite<= sub-neg_binary64 (-.f64 2 (*.f64 6 (*.f64 v v)))))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (/.f64 4 3) -1) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 (-.f64 2 (*.f64 6 (*.f64 v v)))) (PI.f64))) (fma.f64 v v -1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (/.f64 4 3) -1) (*.f64 (*.f64 (sqrt.f64 (-.f64 2 (*.f64 6 (*.f64 v v)))) (PI.f64)) (fma.f64 v v (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (/.f64 4 3) -1) (*.f64 (*.f64 (sqrt.f64 (-.f64 2 (*.f64 6 (*.f64 v v)))) (PI.f64)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 v v) 1)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 4 3) (*.f64 -1 (*.f64 (*.f64 (sqrt.f64 (-.f64 2 (*.f64 6 (*.f64 v v)))) (PI.f64)) (-.f64 (*.f64 v v) 1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 4 3) (Rewrite<= neg-mul-1_binary64 (neg.f64 (*.f64 (*.f64 (sqrt.f64 (-.f64 2 (*.f64 6 (*.f64 v v)))) (PI.f64)) (-.f64 (*.f64 v v) 1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 4 3) (Rewrite<= distribute-rgt-neg-out_binary64 (*.f64 (*.f64 (sqrt.f64 (-.f64 2 (*.f64 6 (*.f64 v v)))) (PI.f64)) (neg.f64 (-.f64 (*.f64 v v) 1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 4 3) (*.f64 (*.f64 (sqrt.f64 (-.f64 2 (*.f64 6 (*.f64 v v)))) (PI.f64)) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (*.f64 v v) 1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 4 3) (*.f64 (*.f64 (sqrt.f64 (-.f64 2 (*.f64 6 (*.f64 v v)))) (PI.f64)) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (*.f64 v v)) 1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 4 3) (*.f64 (*.f64 (sqrt.f64 (-.f64 2 (*.f64 6 (*.f64 v v)))) (PI.f64)) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (*.f64 v v))) 1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 4 3) (*.f64 (*.f64 (sqrt.f64 (-.f64 2 (*.f64 6 (*.f64 v v)))) (PI.f64)) (Rewrite<= +-commutative_binary64 (+.f64 1 (neg.f64 (*.f64 v v)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 4 3) (*.f64 (*.f64 (sqrt.f64 (-.f64 2 (*.f64 6 (*.f64 v v)))) (PI.f64)) (Rewrite<= sub-neg_binary64 (-.f64 1 (*.f64 v v))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 4 3) (Rewrite<= associate-*r*_binary64 (*.f64 (sqrt.f64 (-.f64 2 (*.f64 6 (*.f64 v v)))) (*.f64 (PI.f64) (-.f64 1 (*.f64 v v)))))): 0 points increase in error, 3 points decrease in error
   (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (/.f64 4 3) (*.f64 (PI.f64) (-.f64 1 (*.f64 v v)))) (sqrt.f64 (-.f64 2 (*.f64 6 (*.f64 v v)))))): 1 points increase in error, 1 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 6 (*.f64 v v))))): 1 points increase in error, 1 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 6 (*.f64 v v))))): 2 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))))))): 247 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	26368

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

Alternative 2
Error	1.0
Cost	13952

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

Alternative 3
Error	0.6
Cost	13504

\[\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {2}^{-0.5} \]

Alternative 4
Error	0.7
Cost	13056

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

Reproduce

herbie shell --seed 2022313 
(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)))))))