Falkner and Boettcher, Equation (20:1,3)

Average Error: 0.5 → 0.3

Time: 10.1s

Precision: binary64

Cost: 20608

Math

\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

↓

\[\frac{\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi}}{t}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(1 - v \cdot v\right)} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))

↓

(FPCore (v t)
 :precision binary64
 (/
  (/ (/ (fma v (* v -5.0) 1.0) PI) t)
  (* (sqrt (+ 2.0 (* (* v v) -6.0))) (- 1.0 (* v v)))))

C

double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / (((((double) M_PI) * t) * sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)));
}

↓

double code(double v, double t) {
	return ((fma(v, (v * -5.0), 1.0) / ((double) M_PI)) / t) / (sqrt((2.0 + ((v * v) * -6.0))) * (1.0 - (v * v)));
}

Julia

function code(v, t)
	return Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(Float64(pi * t) * sqrt(Float64(2.0 * Float64(1.0 - Float64(3.0 * Float64(v * v)))))) * Float64(1.0 - Float64(v * v))))
end

↓

function code(v, t)
	return Float64(Float64(Float64(fma(v, Float64(v * -5.0), 1.0) / pi) / t) / Float64(sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0))) * Float64(1.0 - Float64(v * v))))
end

Wolfram

code[v_, t_] := N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * t), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(1.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[v_, t_] := N[(N[(N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / Pi), $MachinePrecision] / t), $MachinePrecision] / N[(N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.5

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

2. Simplified 0.4

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(1 - v \cdot v\right)}} \]
   Proof
   (/.f64 (/.f64 (fma.f64 (*.f64 v v) -5 1) (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (+.f64 2 (*.f64 (*.f64 v v) -6))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (fma.f64 (*.f64 v v) (Rewrite<= metadata-eval (neg.f64 5)) 1) (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (+.f64 2 (*.f64 (*.f64 v v) -6))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 v v) (neg.f64 5)) 1)) (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (+.f64 2 (*.f64 (*.f64 v v) -6))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 v v) 5))) 1) (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (+.f64 2 (*.f64 (*.f64 v v) -6))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 5 (*.f64 v v)))) 1) (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (+.f64 2 (*.f64 (*.f64 v v) -6))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (neg.f64 (*.f64 5 (*.f64 v v))))) (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (+.f64 2 (*.f64 (*.f64 v v) -6))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 (*.f64 5 (*.f64 v v)))) (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (+.f64 2 (*.f64 (*.f64 v v) -6))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 1 2)) (*.f64 (*.f64 v v) -6))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (+.f64 (*.f64 1 2) (*.f64 (*.f64 v v) (Rewrite<= metadata-eval (*.f64 -3 2))))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (+.f64 (*.f64 1 2) (*.f64 (*.f64 v v) (*.f64 (Rewrite<= metadata-eval (neg.f64 3)) 2)))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (+.f64 (*.f64 1 2) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 v v) (neg.f64 3)) 2)))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (+.f64 (*.f64 1 2) (*.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 v v) 3))) 2))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (+.f64 (*.f64 1 2) (*.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 v v)))) 2))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 2 (+.f64 1 (neg.f64 (*.f64 3 (*.f64 v v))))))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (*.f64 (PI.f64) t)) (*.f64 (sqrt.f64 (*.f64 2 (Rewrite<= sub-neg_binary64 (-.f64 1 (*.f64 3 (*.f64 v v)))))) (-.f64 1 (*.f64 v v)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r*_binary64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (*.f64 (*.f64 (PI.f64) t) (*.f64 (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v))))) (-.f64 1 (*.f64 v v)))))): 46 points increase in error, 46 points decrease in error
   (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (PI.f64) t) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))) (-.f64 1 (*.f64 v v))))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.4

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi} \cdot \frac{1}{t}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(1 - v \cdot v\right)} \]

4. Applied egg-rr 0.3

   \[\leadsto \frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi}}{t}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(1 - v \cdot v\right)} \]

5. Final simplification 0.3

   \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi}}{t}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(1 - v \cdot v\right)} \]

Alternatives

Alternative 1
Error	0.4
Cost	14464

\[\frac{1 + -5 \cdot \left(v \cdot v\right)}{t \cdot \left(\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 \cdot \left(1 + \left(v \cdot v\right) \cdot -3\right)}\right)\right)} \]

Alternative 2
Error	0.4
Cost	14336

\[\frac{\frac{1 + -5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(1 - v \cdot v\right)} \]

Alternative 3
Error	1.1
Cost	13312

\[\frac{1}{\frac{t \cdot \sqrt{2}}{\frac{1}{\pi}}} \]

Alternative 4
Error	1.1
Cost	13184

\[\frac{1}{\left(\pi \cdot t\right) \cdot \sqrt{2}} \]

Alternative 5
Error	1.0
Cost	13184

\[\frac{\frac{1}{\pi \cdot t}}{\sqrt{2}} \]

Alternative 6
Error	1.3
Cost	13056

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

Alternative 7
Error	1.3
Cost	13056

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

Reproduce

herbie shell --seed 2022330 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))