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

Average Error: 0.4 → 0.4

Time: 10.9s

Precision: binary64

Cost: 14336

Math

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

↓

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

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

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

Java

public static double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / (((Math.PI * t) * Math.sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)));
}

↓

public static double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / (Math.PI * ((1.0 - (v * v)) * (t * Math.sqrt((2.0 - (v * (v * 6.0)))))));
}

Python

def code(v, t):
	return (1.0 - (5.0 * (v * v))) / (((math.pi * t) * math.sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)))

↓

def code(v, t):
	return (1.0 - (5.0 * (v * v))) / (math.pi * ((1.0 - (v * v)) * (t * math.sqrt((2.0 - (v * (v * 6.0)))))))

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(5.0 * Float64(v * v))) / Float64(pi * Float64(Float64(1.0 - Float64(v * v)) * Float64(t * sqrt(Float64(2.0 - Float64(v * Float64(v * 6.0))))))))
end

MATLAB

function tmp = code(v, t)
	tmp = (1.0 - (5.0 * (v * v))) / (((pi * t) * sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)));
end

↓

function tmp = code(v, t)
	tmp = (1.0 - (5.0 * (v * v))) / (pi * ((1.0 - (v * v)) * (t * sqrt((2.0 - (v * (v * 6.0)))))));
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[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(t * N[Sqrt[N[(2.0 - N[(v * N[(v * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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

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

   [Start] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(1 - v \cdot v\right) \cdot \left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(1 - v \cdot v\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\pi \cdot t\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(1 - v \cdot v\right) \cdot \color{blue}{\left(\pi \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot t\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot t\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \color{blue}{\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-13 [=>] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \left(t \cdot \sqrt{\color{blue}{1 \cdot 2 - 2 \cdot \left(3 \cdot \left(v \cdot v\right)\right)}}\right)\right)} \]
   metadata-eval [=>] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \left(t \cdot \sqrt{\color{blue}{2} - 2 \cdot \left(3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \left(t \cdot \sqrt{2 - 2 \cdot \color{blue}{\left(v \cdot \left(3 \cdot v\right)\right)}}\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \left(t \cdot \sqrt{2 - \color{blue}{v \cdot \left(2 \cdot \left(3 \cdot v\right)\right)}}\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \left(t \cdot \sqrt{2 - v \cdot \left(2 \cdot \color{blue}{\left(v \cdot 3\right)}\right)}\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \left(t \cdot \sqrt{2 - v \cdot \color{blue}{\left(v \cdot \left(2 \cdot 3\right)\right)}}\right)\right)} \]
   metadata-eval [=>] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \left(t \cdot \sqrt{2 - v \cdot \left(v \cdot \color{blue}{6}\right)}\right)\right)} \]

3. Final simplification 0.4

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

Alternatives

Alternative 1
Error	1.0
Cost	13952

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

Alternative 2
Error	1.1
Cost	13568

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

Alternative 3
Error	1.1
Cost	13184

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

Alternative 4
Error	1.4
Cost	13056

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

Reproduce

herbie shell --seed 2023090 
(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)))))