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

Percentage Accurate: 99.3% → 99.4%

Time: 11.8s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

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)))))

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)));
}

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)));
}

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)))

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

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

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]

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

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)))))

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)));
}

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)));
}

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)))

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

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

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]

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1 - v \cdot \left(v \cdot 5\right)}{t \cdot \left(\left(\pi \cdot \left(1 - {v}^{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left({v}^{2}, -6, 2\right)}\right)} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* v (* v 5.0)))
  (* t (* (* PI (- 1.0 (pow v 2.0))) (sqrt (fma (pow v 2.0) -6.0 2.0))))))

C

double code(double v, double t) {
	return (1.0 - (v * (v * 5.0))) / (t * ((((double) M_PI) * (1.0 - pow(v, 2.0))) * sqrt(fma(pow(v, 2.0), -6.0, 2.0))));
}

Julia

function code(v, t)
	return Float64(Float64(1.0 - Float64(v * Float64(v * 5.0))) / Float64(t * Float64(Float64(pi * Float64(1.0 - (v ^ 2.0))) * sqrt(fma((v ^ 2.0), -6.0, 2.0)))))
end

Wolfram

code[v_, t_] := N[(N[(1.0 - N[(v * N[(v * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[(N[(Pi * N[(1.0 - N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Power[v, 2.0], $MachinePrecision] * -6.0 + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in t around 0 99.3%

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

   1. associate-*l* 99.6%

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

   2. +-commutative 99.6%

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

   3. *-commutative 99.6%

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

   4. fma-undefine 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 99.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (/ (fma (* v v) -5.0 1.0) (sqrt (+ 2.0 (* -6.0 (* v v)))))
  (* PI (* t (fma v (- v) 1.0)))))

C

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

Julia

function code(v, t)
	return Float64(Float64(fma(Float64(v * v), -5.0, 1.0) / sqrt(Float64(2.0 + Float64(-6.0 * Float64(v * v))))) / Float64(pi * Float64(t * fma(v, Float64(-v), 1.0))))
end

Wolfram

code[v_, t_] := N[(N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[Sqrt[N[(2.0 + N[(-6.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(t * N[(v * (-v) + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (* (- 1.0 (* v v)) (* (sqrt (* 2.0 (- 1.0 (* (* v v) 3.0)))) (* t PI)))))

C

double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / ((1.0 - (v * v)) * (sqrt((2.0 * (1.0 - ((v * v) * 3.0)))) * (t * ((double) M_PI))));
}

Java

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

Python

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

Julia

function code(v, t)
	return Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(1.0 - Float64(v * v)) * Float64(sqrt(Float64(2.0 * Float64(1.0 - Float64(Float64(v * v) * 3.0)))) * Float64(t * pi))))
end

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(1.0 - N[(N[(v * v), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Final simplification 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* v (* v 5.0)))
  (* (sqrt (+ 2.0 (* 2.0 (* (* v v) -3.0)))) (* (- 1.0 (* v v)) (* t PI)))))

C

double code(double v, double t) {
	return (1.0 - (v * (v * 5.0))) / (sqrt((2.0 + (2.0 * ((v * v) * -3.0)))) * ((1.0 - (v * v)) * (t * ((double) M_PI))));
}

Java

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

Python

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

Julia

function code(v, t)
	return Float64(Float64(1.0 - Float64(v * Float64(v * 5.0))) / Float64(sqrt(Float64(2.0 + Float64(2.0 * Float64(Float64(v * v) * -3.0)))) * Float64(Float64(1.0 - Float64(v * v)) * Float64(t * pi))))
end

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1.0 - N[(v * N[(v * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(2.0 + N[(2.0 * N[(N[(v * v), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(t * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 5: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{t} \cdot \frac{1}{\frac{\pi}{\sqrt{0.5}}} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (* (/ 1.0 t) (/ 1.0 (/ PI (sqrt 0.5)))))

C

double code(double v, double t) {
	return (1.0 / t) * (1.0 / (((double) M_PI) / sqrt(0.5)));
}

Java

public static double code(double v, double t) {
	return (1.0 / t) * (1.0 / (Math.PI / Math.sqrt(0.5)));
}

Python

def code(v, t):
	return (1.0 / t) * (1.0 / (math.pi / math.sqrt(0.5)))

Julia

function code(v, t)
	return Float64(Float64(1.0 / t) * Float64(1.0 / Float64(pi / sqrt(0.5))))
end

MATLAB

function tmp = code(v, t)
	tmp = (1.0 / t) * (1.0 / (pi / sqrt(0.5)));
end

Wolfram

code[v_, t_] := N[(N[(1.0 / t), $MachinePrecision] * N[(1.0 / N[(Pi / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in v around 0 98.1%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 98.1%

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

   2. associate-/r* 98.1%

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

7. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{1 \cdot \frac{\frac{\sqrt{0.5}}{t}}{\pi}} \]
8. Step-by-step derivation

   1. associate-/l/ 98.1%

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

   2. *-commutative 98.1%

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

   3. clear-num 98.2%

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

   4. associate-*r/ 98.7%

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

   5. inv-pow 98.7%

      \[\leadsto 1 \cdot \color{blue}{{\left(t \cdot \frac{\pi}{\sqrt{0.5}}\right)}^{-1}} \]

   6. unpow-prod-down 98.7%

      \[\leadsto 1 \cdot \color{blue}{\left({t}^{-1} \cdot {\left(\frac{\pi}{\sqrt{0.5}}\right)}^{-1}\right)} \]

   7. inv-pow 98.7%

      \[\leadsto 1 \cdot \left(\color{blue}{\frac{1}{t}} \cdot {\left(\frac{\pi}{\sqrt{0.5}}\right)}^{-1}\right) \]

9. Applied egg-rr 98.7%

   \[\leadsto 1 \cdot \color{blue}{\left(\frac{1}{t} \cdot {\left(\frac{\pi}{\sqrt{0.5}}\right)}^{-1}\right)} \]
10. Step-by-step derivation

    1. unpow-1 98.7%

       \[\leadsto 1 \cdot \left(\frac{1}{t} \cdot \color{blue}{\frac{1}{\frac{\pi}{\sqrt{0.5}}}}\right) \]

11. Simplified 98.7%

    \[\leadsto 1 \cdot \color{blue}{\left(\frac{1}{t} \cdot \frac{1}{\frac{\pi}{\sqrt{0.5}}}\right)} \]

12. Final simplification 98.7%

    \[\leadsto \frac{1}{t} \cdot \frac{1}{\frac{\pi}{\sqrt{0.5}}} \]
13. Add Preprocessing

Alternative 6: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5} \cdot \frac{1}{t \cdot \pi} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (* (sqrt 0.5) (/ 1.0 (* t PI))))

C

double code(double v, double t) {
	return sqrt(0.5) * (1.0 / (t * ((double) M_PI)));
}

Java

public static double code(double v, double t) {
	return Math.sqrt(0.5) * (1.0 / (t * Math.PI));
}

Python

def code(v, t):
	return math.sqrt(0.5) * (1.0 / (t * math.pi))

Julia

function code(v, t)
	return Float64(sqrt(0.5) * Float64(1.0 / Float64(t * pi)))
end

MATLAB

function tmp = code(v, t)
	tmp = sqrt(0.5) * (1.0 / (t * pi));
end

Wolfram

code[v_, t_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(1.0 / N[(t * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in v around 0 98.1%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]
6. Step-by-step derivation

   1. div-inv 98.1%

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

7. Applied egg-rr 98.1%

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

8. Final simplification 98.1%

   \[\leadsto \sqrt{0.5} \cdot \frac{1}{t \cdot \pi} \]
9. Add Preprocessing

Alternative 7: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{t \cdot \frac{\pi}{\sqrt{0.5}}} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ 1.0 (* t (/ PI (sqrt 0.5)))))

C

double code(double v, double t) {
	return 1.0 / (t * (((double) M_PI) / sqrt(0.5)));
}

Java

public static double code(double v, double t) {
	return 1.0 / (t * (Math.PI / Math.sqrt(0.5)));
}

Python

def code(v, t):
	return 1.0 / (t * (math.pi / math.sqrt(0.5)))

Julia

function code(v, t)
	return Float64(1.0 / Float64(t * Float64(pi / sqrt(0.5))))
end

MATLAB

function tmp = code(v, t)
	tmp = 1.0 / (t * (pi / sqrt(0.5)));
end

Wolfram

code[v_, t_] := N[(1.0 / N[(t * N[(Pi / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in v around 0 98.1%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]
6. Step-by-step derivation

   1. clear-num 98.2%

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

   2. inv-pow 98.2%

      \[\leadsto \color{blue}{{\left(\frac{t \cdot \pi}{\sqrt{0.5}}\right)}^{-1}} \]

7. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{{\left(\frac{t \cdot \pi}{\sqrt{0.5}}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 98.2%

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

   2. associate-/l* 98.7%

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

9. Simplified 98.7%

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

10. Final simplification 98.7%

    \[\leadsto \frac{1}{t \cdot \frac{\pi}{\sqrt{0.5}}} \]
11. Add Preprocessing

Alternative 8: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{0.5}}{t \cdot \pi} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ (sqrt 0.5) (* t PI)))

C

double code(double v, double t) {
	return sqrt(0.5) / (t * ((double) M_PI));
}

Java

public static double code(double v, double t) {
	return Math.sqrt(0.5) / (t * Math.PI);
}

Python

def code(v, t):
	return math.sqrt(0.5) / (t * math.pi)

Julia

function code(v, t)
	return Float64(sqrt(0.5) / Float64(t * pi))
end

MATLAB

function tmp = code(v, t)
	tmp = sqrt(0.5) / (t * pi);
end

Wolfram

code[v_, t_] := N[(N[Sqrt[0.5], $MachinePrecision] / N[(t * Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in v around 0 98.1%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]

6. Final simplification 98.1%

   \[\leadsto \frac{\sqrt{0.5}}{t \cdot \pi} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024072 
(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)))))