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

Percentage Accurate: 99.3% → 99.5%

Time: 11.1s

Alternatives: 10

Speedup: 1.2×

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

4. Applied egg-rr 99.6%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

4. Applied egg-rr 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 3: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

4. Applied egg-rr 99.1%

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

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

Alternative 4: 98.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 97.3

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

5. Simplified 97.3%

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

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 97.8

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

7. Applied egg-rr 97.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l/ N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 98.1

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

9. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{\frac{\frac{1}{\pi \cdot \sqrt{2}}}{t}} \]
10. Add Preprocessing

Alternative 5: 98.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 97.3

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

5. Simplified 97.3%

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

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 97.8

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

7. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\frac{\frac{1}{\pi}}{t \cdot \sqrt{2}}} \]
8. Add Preprocessing

Alternative 6: 98.4% accurate, 2.4× 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 99.1%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 97.3

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

5. Simplified 97.3%

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

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 97.4

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

7. Applied egg-rr 97.4%

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

8. Final simplification 97.4%

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

Alternative 7: 98.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 97.3

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

5. Simplified 97.3%

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

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. lower-*.f64 97.3

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 97.3

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

7. Applied egg-rr 97.3%

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

8. Final simplification 97.3%

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

Alternative 8: 98.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 97.3

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

5. Simplified 97.3%

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

Alternative 9: 97.8% accurate, 2.4× speedup

Math

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

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

Derivation

1. Initial program 99.1%

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 97.2

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

7. Simplified 97.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l/ N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. associate-/l/ N/A

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

   12. lift-*.f64 N/A

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

   13. lower-/.f64 97.2

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

9. Applied egg-rr 97.2%

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

Alternative 10: 97.8% accurate, 2.8× 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 99.1%

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 97.2

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

7. Simplified 97.2%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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)))))