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

Average Error: 0.5 → 0.4

Time: 9.2s

Precision: binary64

Cost: 20864

Math

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

↓

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

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 (* t (* PI (sqrt (* 2.0 (fma (* v v) -3.0 1.0))))))
  (/ (+ 1.0 (* (* v v) -5.0)) (- 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 (1.0 / (t * (((double) M_PI) * sqrt((2.0 * fma((v * v), -3.0, 1.0)))))) * ((1.0 + ((v * v) * -5.0)) / (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(1.0 / Float64(t * Float64(pi * sqrt(Float64(2.0 * fma(Float64(v * v), -3.0, 1.0)))))) * Float64(Float64(1.0 + Float64(Float64(v * v) * -5.0)) / 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[(1.0 / N[(t * N[(Pi * N[Sqrt[N[(2.0 * N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(v * v), $MachinePrecision] * -5.0), $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. Applied egg-rr 0.4

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

3. Applied egg-rr 0.4

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

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.5
Cost	14464

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

Alternative 2
Error	1.0
Cost	13952

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

Alternative 3
Error	1.0
Cost	13184

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

Alternative 4
Error	1.3
Cost	13056

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

Alternative 5
Error	1.3
Cost	13056

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

Reproduce

herbie shell --seed 2022221 
(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)))))