Average Error: 0.4 → 0.3
Time: 4.3s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\begin{array}{l} t_1 := \pi \cdot \sqrt{2}\\ \frac{1 - 5 \cdot {v}^{2}}{\left(t_1 - {v}^{2} \cdot t_1\right) \cdot t} \cdot \sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}} \end{array} \]
(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
 (let* ((t_1 (* PI (sqrt 2.0))))
   (*
    (/ (- 1.0 (* 5.0 (pow v 2.0))) (* (- t_1 (* (pow v 2.0) t_1)) t))
    (sqrt (/ 1.0 (- 1.0 (* 3.0 (pow v 2.0))))))))
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) {
	double t_1 = ((double) M_PI) * sqrt(2.0);
	return ((1.0 - (5.0 * pow(v, 2.0))) / ((t_1 - (pow(v, 2.0) * t_1)) * t)) * sqrt((1.0 / (1.0 - (3.0 * pow(v, 2.0)))));
}

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) {
	double t_1 = Math.PI * Math.sqrt(2.0);
	return ((1.0 - (5.0 * Math.pow(v, 2.0))) / ((t_1 - (Math.pow(v, 2.0) * t_1)) * t)) * Math.sqrt((1.0 / (1.0 - (3.0 * Math.pow(v, 2.0)))));
}

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):
	t_1 = math.pi * math.sqrt(2.0)
	return ((1.0 - (5.0 * math.pow(v, 2.0))) / ((t_1 - (math.pow(v, 2.0) * t_1)) * t)) * math.sqrt((1.0 / (1.0 - (3.0 * math.pow(v, 2.0)))))

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)
	t_1 = Float64(pi * sqrt(2.0))
	return Float64(Float64(Float64(1.0 - Float64(5.0 * (v ^ 2.0))) / Float64(Float64(t_1 - Float64((v ^ 2.0) * t_1)) * t)) * sqrt(Float64(1.0 / Float64(1.0 - Float64(3.0 * (v ^ 2.0))))))
end

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)
	t_1 = pi * sqrt(2.0);
	tmp = ((1.0 - (5.0 * (v ^ 2.0))) / ((t_1 - ((v ^ 2.0) * t_1)) * t)) * sqrt((1.0 / (1.0 - (3.0 * (v ^ 2.0)))));
end

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_] := Block[{t$95$1 = N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 - N[(5.0 * N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 - N[(N[Power[v, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(1.0 - N[(3.0 * N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

\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)}
↓
\begin{array}{l}
t_1 := \pi \cdot \sqrt{2}\\
\frac{1 - 5 \cdot {v}^{2}}{\left(t_1 - {v}^{2} \cdot t_1\right) \cdot t} \cdot \sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}}
\end{array}

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Taylor expanded in t around 0 0.3

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

  3. Final simplification0.3

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

Reproduce

herbie shell --seed 2022152 
(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)))))