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

Average Error: 0.4 → 0.1

Time: 12.6s

Precision: binary64

Cost: 20608

Math

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

↓

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

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 (+ 2.0 (* (* v v) -6.0))) (- 1.0 (* v v)))))
  t))

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((2.0 + ((v * v) * -6.0))) * (1.0 - (v * v))))) / t;
}

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 * Float64(sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0))) * Float64(1.0 - Float64(v * v))))) / t)
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[(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]), $MachinePrecision] / t), $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. Applied egg-rr 0.5

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

3. Taylor expanded in t around 0 0.5

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

4. Simplified 0.3

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

   [Start] 0.5	\[ \frac{1}{t \cdot \pi} \cdot \frac{1 + \left(v \cdot v\right) \cdot -5}{\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)} \]
   associate-/l/ [<=] 0.3	\[ \color{blue}{\frac{\frac{1}{\pi}}{t}} \cdot \frac{1 + \left(v \cdot v\right) \cdot -5}{\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)} \]

5. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.3
Cost	14592

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

Alternative 2
Error	0.4
Cost	14464

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

Alternative 3
Error	0.4
Cost	14336

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

Alternative 4
Error	1.3
Cost	13184

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

Alternative 5
Error	1.0
Cost	13184

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

Alternative 6
Error	1.0
Cost	13184

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

Alternative 7
Error	0.7
Cost	13184

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

Alternative 8
Error	1.3
Cost	13056

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

Alternative 9
Error	1.3
Cost	13056

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

Reproduce

herbie shell --seed 2023073 
(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)))))