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

Percentage Accurate: 99.3% → 99.4%

Time: 6.6s

Alternatives: 6

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. associate-*r* 99.4%

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

   2. associate-*r* 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*l* 99.4%

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

   5. expm1-log1p-u 73.0%

      \[\leadsto \frac{1 + \left(v \cdot v\right) \cdot -5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)\right)}} \]

   6. expm1-udef 27.1%

      \[\leadsto \frac{1 + \left(v \cdot v\right) \cdot -5}{\color{blue}{e^{\mathsf{log1p}\left(\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)} - 1}} \]

5. Applied egg-rr 27.1%

   \[\leadsto \frac{1 + \left(v \cdot v\right) \cdot -5}{\color{blue}{e^{\mathsf{log1p}\left(\left(1 - v \cdot v\right) \cdot \left(\left(t \cdot \pi\right) \cdot \sqrt{2 \cdot \left(1 + -3 \cdot \left(v \cdot v\right)\right)}\right)\right)} - 1}} \]
6. Step-by-step derivation

   1. expm1-def 73.0%

      \[\leadsto \frac{1 + \left(v \cdot v\right) \cdot -5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - v \cdot v\right) \cdot \left(\left(t \cdot \pi\right) \cdot \sqrt{2 \cdot \left(1 + -3 \cdot \left(v \cdot v\right)\right)}\right)\right)\right)}} \]

   2. expm1-log1p 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*l* 99.5%

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

   5. associate-*l* 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v, t)
	tmp = (1.0 + ((v * v) * -5.0)) / (t * ((pi * (1.0 - (v * v))) * sqrt((2.0 + ((v * v) * -6.0)))));
end

Wolfram

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

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

   1. associate-*r* 99.4%

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

   2. associate-*r* 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*l* 99.4%

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

   5. expm1-log1p-u 73.0%

      \[\leadsto \frac{1 + \left(v \cdot v\right) \cdot -5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)\right)}} \]

   6. expm1-udef 27.1%

      \[\leadsto \frac{1 + \left(v \cdot v\right) \cdot -5}{\color{blue}{e^{\mathsf{log1p}\left(\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)} - 1}} \]

5. Applied egg-rr 27.1%

   \[\leadsto \frac{1 + \left(v \cdot v\right) \cdot -5}{\color{blue}{e^{\mathsf{log1p}\left(\left(1 - v \cdot v\right) \cdot \left(\left(t \cdot \pi\right) \cdot \sqrt{2 \cdot \left(1 + -3 \cdot \left(v \cdot v\right)\right)}\right)\right)} - 1}} \]
6. Step-by-step derivation

   1. expm1-def 73.0%

      \[\leadsto \frac{1 + \left(v \cdot v\right) \cdot -5}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - v \cdot v\right) \cdot \left(\left(t \cdot \pi\right) \cdot \sqrt{2 \cdot \left(1 + -3 \cdot \left(v \cdot v\right)\right)}\right)\right)\right)}} \]

   2. expm1-log1p 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*l* 99.5%

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

   5. associate-*l* 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in t around 0 99.3%

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

   1. associate-*l* 99.5%

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

   2. unpow2 99.5%

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

   3. *-commutative 99.5%

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

   4. unpow2 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 3: 98.1% accurate, 1.1× 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 99.4%

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

   1. associate-*l* 99.4%

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

   2. associate-/r* 99.4%

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

   3. sub-neg 99.4%

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

   4. +-commutative 99.4%

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

   5. sqr-neg 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-neg-in 99.4%

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

   8. fma-def 99.4%

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

   9. sqr-neg 99.4%

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

   10. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in v around 0 97.6%

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

   1. div-inv 97.6%

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

   2. *-commutative 97.6%

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

6. Applied egg-rr 97.6%

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

7. Final simplification 97.6%

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

Alternative 4: 98.6% accurate, 1.1× 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%

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

4. Taylor expanded in v around 0 98.2%

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

   1. *-commutative 98.2%

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

6. Simplified 98.2%

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

7. Final simplification 98.2%

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

Alternative 5: 99.0% accurate, 1.1× 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 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in v around 0 98.2%

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

   1. associate-/r* 98.1%

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

   2. *-commutative 98.1%

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

6. Simplified 98.1%

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

   1. div-inv 98.2%

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

   2. *-commutative 98.2%

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

8. Applied egg-rr 98.2%

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

   1. associate-*l/ 98.5%

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

   2. *-un-lft-identity 98.5%

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

10. Applied egg-rr 98.5%

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

11. Final simplification 98.5%

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

Alternative 6: 98.0% accurate, 1.1× 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 99.4%

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

   1. associate-*l* 99.4%

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

   2. associate-/r* 99.4%

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

   3. sub-neg 99.4%

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

   4. +-commutative 99.4%

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

   5. sqr-neg 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-neg-in 99.4%

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

   8. fma-def 99.4%

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

   9. sqr-neg 99.4%

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

   10. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in v around 0 97.6%

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

5. Final simplification 97.6%

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

Reproduce

herbie shell --seed 2023292 
(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)))))