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

Average Error: 0.4 → 0.3

Time: 33.0s

Precision: binary64

Cost: 20608

Math

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

↓

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

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
 (/
  (/ (+ 1.0 (* (* v v) -5.0)) t)
  (* (* (- 1.0 (* v v)) (sqrt (fma -6.0 (* v v) 2.0))) PI)))

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

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(1.0 + Float64(Float64(v * v) * -5.0)) / t) / Float64(Float64(Float64(1.0 - Float64(v * v)) * sqrt(fma(-6.0, Float64(v * v), 2.0))) * pi))
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[(1.0 + N[(N[(v * v), $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[(N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-6.0 * N[(v * v), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * Pi), $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))))): 0 points increase in error, 1 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)))))))): 36 points increase in error, 37 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, 1 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. Taylor expanded in t around 0 0.4

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

4. Simplified 0.3

   \[\leadsto \color{blue}{\frac{\frac{1 + \left(v \cdot v\right) \cdot -5}{t}}{\left(\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}\right) \cdot \pi}} \]
   Proof
   (/.f64 (/.f64 (+.f64 1 (*.f64 (*.f64 v v) -5)) t) (*.f64 (*.f64 (-.f64 1 (*.f64 v v)) (sqrt.f64 (fma.f64 -6 (*.f64 v v) 2))) (PI.f64))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 1 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 v 2)) -5)) t) (*.f64 (*.f64 (-.f64 1 (*.f64 v v)) (sqrt.f64 (fma.f64 -6 (*.f64 v v) 2))) (PI.f64))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 1 (Rewrite<= *-commutative_binary64 (*.f64 -5 (pow.f64 v 2)))) t) (*.f64 (*.f64 (-.f64 1 (*.f64 v v)) (sqrt.f64 (fma.f64 -6 (*.f64 v v) 2))) (PI.f64))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 1 (*.f64 -5 (pow.f64 v 2))) t) (*.f64 (*.f64 (-.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 v 2))) (sqrt.f64 (fma.f64 -6 (*.f64 v v) 2))) (PI.f64))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 1 (*.f64 -5 (pow.f64 v 2))) t) (*.f64 (*.f64 (-.f64 1 (pow.f64 v 2)) (sqrt.f64 (fma.f64 -6 (Rewrite<= unpow2_binary64 (pow.f64 v 2)) 2))) (PI.f64))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 1 (*.f64 -5 (pow.f64 v 2))) t) (*.f64 (*.f64 (-.f64 1 (pow.f64 v 2)) (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -6 (pow.f64 v 2)) 2)))) (PI.f64))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 1 (*.f64 -5 (pow.f64 v 2))) t) (Rewrite<= associate-*r*_binary64 (*.f64 (-.f64 1 (pow.f64 v 2)) (*.f64 (sqrt.f64 (+.f64 (*.f64 -6 (pow.f64 v 2)) 2)) (PI.f64))))): 1 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r*_binary64 (/.f64 (+.f64 1 (*.f64 -5 (pow.f64 v 2))) (*.f64 t (*.f64 (-.f64 1 (pow.f64 v 2)) (*.f64 (sqrt.f64 (+.f64 (*.f64 -6 (pow.f64 v 2)) 2)) (PI.f64)))))): 53 points increase in error, 40 points decrease in error

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	14464

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

Alternative 2
Error	0.4
Cost	14336

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

Alternative 3
Error	1.0
Cost	13568

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

Alternative 4
Error	1.1
Cost	13184

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

Alternative 5
Error	1.0
Cost	13184

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

Alternative 6
Error	1.3
Cost	13056

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

Alternative 7
Error	1.3
Cost	13056

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

Reproduce

herbie shell --seed 2022325 
(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)))))