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

Average Error: 0.4 → 0.3

Time: 10.3s

Precision: binary64

Cost: 27008

Math

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

↓

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

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 (sqrt (fma v (* v -6.0) 2.0))))
  (/ 1.0 (* t (- 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) * sqrt(fma(v, (v * -6.0), 2.0)))) * (1.0 / (t * (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(fma(v, Float64(v * -5.0), 1.0) / Float64(pi * sqrt(fma(v, Float64(v * -6.0), 2.0)))) * Float64(1.0 / Float64(t * 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[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(Pi * N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.4

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

2. Simplified 0.4

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\sqrt{\mathsf{fma}\left(v, -6 \cdot v, 2\right)}}}{\pi \cdot \left(t - \left(v \cdot v\right) \cdot t\right)}} \]
   Proof
   (/.f64 (/.f64 (fma.f64 v (*.f64 v -5) 1) (sqrt.f64 (fma.f64 v (*.f64 -6 v) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (fma.f64 v (*.f64 v (Rewrite<= metadata-eval (neg.f64 5))) 1) (sqrt.f64 (fma.f64 v (*.f64 -6 v) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (fma.f64 v (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 v 5))) 1) (sqrt.f64 (fma.f64 v (*.f64 -6 v) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (fma.f64 v (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 5 v))) 1) (sqrt.f64 (fma.f64 v (*.f64 -6 v) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 v (neg.f64 (*.f64 5 v))) 1)) (sqrt.f64 (fma.f64 v (*.f64 -6 v) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 v (*.f64 5 v)))) 1) (sqrt.f64 (fma.f64 v (*.f64 -6 v) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 5 v) v))) 1) (sqrt.f64 (fma.f64 v (*.f64 -6 v) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 (neg.f64 (Rewrite<= associate-*r*_binary64 (*.f64 5 (*.f64 v v)))) 1) (sqrt.f64 (fma.f64 v (*.f64 -6 v) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (neg.f64 (*.f64 5 (*.f64 v v))))) (sqrt.f64 (fma.f64 v (*.f64 -6 v) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 (*.f64 5 (*.f64 v v)))) (sqrt.f64 (fma.f64 v (*.f64 -6 v) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (fma.f64 v (*.f64 (Rewrite<= metadata-eval (*.f64 2 -3)) v) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (fma.f64 v (*.f64 (*.f64 2 (Rewrite<= metadata-eval (neg.f64 3))) v) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (fma.f64 v (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 (neg.f64 3) v))) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (fma.f64 v (*.f64 2 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 3 v)))) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 v (*.f64 2 (neg.f64 (*.f64 3 v)))) 2)))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 (neg.f64 (*.f64 3 v))) v)) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 (neg.f64 (*.f64 3 v)) v))) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (+.f64 (*.f64 2 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (*.f64 3 v) v)))) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (+.f64 (*.f64 2 (neg.f64 (Rewrite<= associate-*r*_binary64 (*.f64 3 (*.f64 v v))))) 2))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (+.f64 (*.f64 2 (neg.f64 (*.f64 3 (*.f64 v v)))) (Rewrite<= metadata-eval (*.f64 2 1))))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 2 (+.f64 (neg.f64 (*.f64 3 (*.f64 v v))) 1))))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 1 (neg.f64 (*.f64 3 (*.f64 v v)))))))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (*.f64 2 (Rewrite<= sub-neg_binary64 (-.f64 1 (*.f64 3 (*.f64 v v))))))) (*.f64 (PI.f64) (-.f64 t (*.f64 (*.f64 v v) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))) (*.f64 (PI.f64) (Rewrite<= unsub-neg_binary64 (+.f64 t (neg.f64 (*.f64 (*.f64 v v) t)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))) (*.f64 (PI.f64) (+.f64 t (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (*.f64 v v)) t))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))) (*.f64 (PI.f64) (+.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 t)) (*.f64 (neg.f64 (*.f64 v v)) t)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))) (*.f64 (PI.f64) (Rewrite<= distribute-rgt-in_binary64 (*.f64 t (+.f64 1 (neg.f64 (*.f64 v v))))))): 1 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))) (*.f64 (PI.f64) (*.f64 t (Rewrite<= sub-neg_binary64 (-.f64 1 (*.f64 v v)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (PI.f64) t) (-.f64 1 (*.f64 v v))))): 2 points increase in error, 2 points decrease in error
   (/.f64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))) (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 1 (*.f64 v v)) (*.f64 (PI.f64) t)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-/l/_binary64 (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (*.f64 (*.f64 (-.f64 1 (*.f64 v v)) (*.f64 (PI.f64) t)) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v)))))))): 27 points increase in error, 36 points decrease in error
   (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (Rewrite<= associate-*r*_binary64 (*.f64 (-.f64 1 (*.f64 v v)) (*.f64 (*.f64 (PI.f64) t) (sqrt.f64 (*.f64 2 (-.f64 1 (*.f64 3 (*.f64 v v))))))))): 0 points increase in error, 2 points decrease in error
   (/.f64 (-.f64 1 (*.f64 5 (*.f64 v v))) (Rewrite<= *-commutative_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 0.3

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

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	20992

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

Alternative 2
Error	0.4
Cost	14464

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

Alternative 3
Error	0.8
Cost	13184

\[\frac{\frac{1}{\frac{\pi}{\sqrt{0.5}}}}{t} \]

Alternative 4
Error	1.3
Cost	13056

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

Alternative 5
Error	1.3
Cost	13056

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

Alternative 6
Error	1.4
Cost	13056

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

Reproduce

herbie shell --seed 2022310 
(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)))))