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

Percentage Accurate: 99.3% → 99.4%

Time: 9.7s

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, 0.4× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (/ (fma (* v v) -5.0 1.0) (sqrt (+ 2.0 (* (* v v) -6.0))))
  (* 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 + ((v * v) * -6.0)))) / (((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(Float64(v * v) * -6.0)))) / 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[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(t * N[(v * (-v) + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 99.3%

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

6. Taylor expanded in v around 0 99.3%

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

   1. inv-pow 99.3%

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

   2. *-commutative 99.3%

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

   3. unpow-prod-down 99.5%

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

   4. inv-pow 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

   \[\leadsto {\left(\pi \cdot \sqrt{2}\right)}^{-1} \cdot \frac{1}{t} \]
10. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 4: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\pi}}{t \cdot {2}^{0.5}} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ (/ 1.0 PI) (* t (pow 2.0 0.5))))

C

double code(double v, double t) {
	return (1.0 / ((double) M_PI)) / (t * pow(2.0, 0.5));
}

Java

public static double code(double v, double t) {
	return (1.0 / Math.PI) / (t * Math.pow(2.0, 0.5));
}

Python

def code(v, t):
	return (1.0 / math.pi) / (t * math.pow(2.0, 0.5))

Julia

function code(v, t)
	return Float64(Float64(1.0 / pi) / Float64(t * (2.0 ^ 0.5)))
end

MATLAB

function tmp = code(v, t)
	tmp = (1.0 / pi) / (t * (2.0 ^ 0.5));
end

Wolfram

code[v_, t_] := N[(N[(1.0 / Pi), $MachinePrecision] / N[(t * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 99.3%

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

6. Taylor expanded in v around 0 99.3%

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

   1. inv-pow 99.3%

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

   2. *-commutative 99.3%

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

   3. unpow-prod-down 99.5%

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

   4. inv-pow 99.5%

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

8. Applied egg-rr 99.5%

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

   1. *-commutative 99.5%

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

   2. unpow-1 99.5%

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

   3. div-inv 99.4%

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

   4. associate-/r* 99.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{t}}{\pi}}{\sqrt{2}}} \]

   5. associate-/r* 99.2%

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

   6. add-sqr-sqrt 98.6%

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

   7. associate-/r* 98.8%

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

   8. *-commutative 98.8%

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

   9. pow1/2 98.8%

      \[\leadsto \frac{\frac{\frac{1}{\pi \cdot t}}{\sqrt{\color{blue}{{2}^{0.5}}}}}{\sqrt{\sqrt{2}}} \]

   10. sqrt-pow1 98.8%

       \[\leadsto \frac{\frac{\frac{1}{\pi \cdot t}}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}}{\sqrt{\sqrt{2}}} \]

   11. metadata-eval 98.8%

       \[\leadsto \frac{\frac{\frac{1}{\pi \cdot t}}{{2}^{\color{blue}{0.25}}}}{\sqrt{\sqrt{2}}} \]

   12. pow1/2 98.8%

       \[\leadsto \frac{\frac{\frac{1}{\pi \cdot t}}{{2}^{0.25}}}{\sqrt{\color{blue}{{2}^{0.5}}}} \]

   13. sqrt-pow1 98.8%

       \[\leadsto \frac{\frac{\frac{1}{\pi \cdot t}}{{2}^{0.25}}}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}} \]

   14. metadata-eval 98.8%

       \[\leadsto \frac{\frac{\frac{1}{\pi \cdot t}}{{2}^{0.25}}}{{2}^{\color{blue}{0.25}}} \]

10. Applied egg-rr 98.8%

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{\pi \cdot t}}{{2}^{0.25}}}{{2}^{0.25}}} \]
11. Step-by-step derivation

    1. associate-/l/ 98.6%

       \[\leadsto \color{blue}{\frac{\frac{1}{\pi \cdot t}}{{2}^{0.25} \cdot {2}^{0.25}}} \]

    2. associate-/r* 98.3%

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\pi}}{t}}}{{2}^{0.25} \cdot {2}^{0.25}} \]

    3. associate-/l/ 98.4%

       \[\leadsto \color{blue}{\frac{\frac{1}{\pi}}{\left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot t}} \]

    4. pow-sqr 99.4%

       \[\leadsto \frac{\frac{1}{\pi}}{\color{blue}{{2}^{\left(2 \cdot 0.25\right)}} \cdot t} \]

    5. metadata-eval 99.4%

       \[\leadsto \frac{\frac{1}{\pi}}{{2}^{\color{blue}{0.5}} \cdot t} \]

12. Simplified 99.4%

    \[\leadsto \color{blue}{\frac{\frac{1}{\pi}}{{2}^{0.5} \cdot t}} \]

13. Final simplification 99.4%

    \[\leadsto \frac{\frac{1}{\pi}}{t \cdot {2}^{0.5}} \]
14. Add Preprocessing

Alternative 5: 98.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 99.3%

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

6. Taylor expanded in v around 0 99.3%

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

7. Final simplification 99.3%

   \[\leadsto \frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)} \]
8. Add Preprocessing

Alternative 6: 98.5% accurate, 1.2× speedup

Math

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

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(Float64(1.0 / 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[(N[(1.0 / t), $MachinePrecision] / N[(Pi / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 98.8%

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

   1. add-sqr-sqrt 43.5%

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

   2. sqrt-unprod 27.7%

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

   3. frac-times 27.4%

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

   4. rem-square-sqrt 27.6%

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

   5. pow2 27.6%

      \[\leadsto \sqrt{\frac{0.5}{\color{blue}{{\left(t \cdot \pi\right)}^{2}}}} \]

   6. *-commutative 27.6%

      \[\leadsto \sqrt{\frac{0.5}{{\color{blue}{\left(\pi \cdot t\right)}}^{2}}} \]

7. Applied egg-rr 27.6%

   \[\leadsto \color{blue}{\sqrt{\frac{0.5}{{\left(\pi \cdot t\right)}^{2}}}} \]
8. Step-by-step derivation

   1. sqrt-div 27.6%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\sqrt{{\left(\pi \cdot t\right)}^{2}}}} \]

   2. *-commutative 27.6%

      \[\leadsto \frac{\sqrt{0.5}}{\sqrt{{\color{blue}{\left(t \cdot \pi\right)}}^{2}}} \]

   3. sqrt-pow1 98.8%

      \[\leadsto \frac{\sqrt{0.5}}{\color{blue}{{\left(t \cdot \pi\right)}^{\left(\frac{2}{2}\right)}}} \]

   4. metadata-eval 98.8%

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

   5. pow1 98.8%

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

   6. *-commutative 98.8%

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

   7. un-div-inv 98.8%

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

   8. associate-*r/ 98.8%

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

   9. times-frac 98.7%

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

9. Applied egg-rr 98.7%

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

    1. *-commutative 98.7%

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

    2. clear-num 99.5%

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

    3. un-div-inv 99.4%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

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

Alternative 7: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 98.8%

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

6. Final simplification 98.8%

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

Alternative 8: 98.0% accurate, 1.2× speedup

Math

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

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(Float64(sqrt(0.5) / t) / pi)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 98.8%

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

   1. add-sqr-sqrt 43.5%

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

   2. sqrt-unprod 27.7%

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

   3. frac-times 27.4%

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

   4. rem-square-sqrt 27.6%

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

   5. pow2 27.6%

      \[\leadsto \sqrt{\frac{0.5}{\color{blue}{{\left(t \cdot \pi\right)}^{2}}}} \]

   6. *-commutative 27.6%

      \[\leadsto \sqrt{\frac{0.5}{{\color{blue}{\left(\pi \cdot t\right)}}^{2}}} \]

7. Applied egg-rr 27.6%

   \[\leadsto \color{blue}{\sqrt{\frac{0.5}{{\left(\pi \cdot t\right)}^{2}}}} \]
8. Step-by-step derivation

   1. sqrt-div 27.6%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\sqrt{{\left(\pi \cdot t\right)}^{2}}}} \]

   2. *-commutative 27.6%

      \[\leadsto \frac{\sqrt{0.5}}{\sqrt{{\color{blue}{\left(t \cdot \pi\right)}}^{2}}} \]

   3. sqrt-pow1 98.8%

      \[\leadsto \frac{\sqrt{0.5}}{\color{blue}{{\left(t \cdot \pi\right)}^{\left(\frac{2}{2}\right)}}} \]

   4. metadata-eval 98.8%

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

   5. pow1 98.8%

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

   6. *-commutative 98.8%

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

   7. un-div-inv 98.8%

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

   8. associate-*r/ 98.8%

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

   9. times-frac 98.7%

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

9. Applied egg-rr 98.7%

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

    1. associate-*l/ 98.8%

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

    2. un-div-inv 98.9%

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

11. Applied egg-rr 98.9%

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

12. Final simplification 98.9%

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

Reproduce

herbie shell --seed 2024040 
(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)))))