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

Percentage Accurate: 99.3% → 99.6%

Time: 10.7s

Alternatives: 7

Speedup: 0.7×

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: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. associate-*l* 99.1%

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

   2. associate-*l* 99.1%

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

   3. associate-/r* 99.6%

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

   4. cancel-sign-sub-inv 99.6%

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

   5. sqr-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. *-commutative 99.6%

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

   8. fma-def 99.6%

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

   9. sqr-neg 99.6%

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

   10. metadata-eval 99.6%

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

   11. associate-*r* 99.6%

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

   12. *-commutative 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 98.3% 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 99.1%

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

   1. associate-*r* 99.1%

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

   2. associate-*l* 99.1%

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

   3. cancel-sign-sub-inv 99.1%

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

   4. sqr-neg 99.1%

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

   5. metadata-eval 99.1%

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

   6. sqr-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 98.7%

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

5. Final simplification 98.7%

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

Alternative 3: 98.4% accurate, 1.1× 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 99.1%

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

   1. associate-*l* 99.1%

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

   2. associate-*l* 99.1%

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

   3. associate-/r* 99.6%

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

   4. cancel-sign-sub-inv 99.6%

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

   5. sqr-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. *-commutative 99.6%

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

   8. fma-def 99.6%

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

   9. sqr-neg 99.6%

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

   10. metadata-eval 99.6%

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

   11. associate-*r* 99.6%

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

   12. *-commutative 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 98.1%

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

   1. associate-/r* 98.4%

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

   2. div-inv 98.2%

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

6. Applied egg-rr 98.2%

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

   1. associate-*l/ 98.4%

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

   2. div-inv 98.4%

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

   3. div-inv 98.3%

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

   4. clear-num 99.1%

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

   5. associate-*l/ 99.0%

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

   6. div-inv 99.0%

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 4: 98.5% accurate, 1.1× 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. Step-by-step derivation

   1. associate-*r* 99.1%

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

   2. associate-*l* 99.1%

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

   3. cancel-sign-sub-inv 99.1%

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

   4. sqr-neg 99.1%

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

   5. metadata-eval 99.1%

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

   6. sqr-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 98.7%

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

   1. associate-*l* 98.7%

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

   2. unpow2 98.7%

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

   3. cancel-sign-sub-inv 98.7%

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

   4. metadata-eval 98.7%

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

   5. *-un-lft-identity 98.7%

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

   6. +-commutative 98.7%

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

   7. *-commutative 98.7%

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

   8. fma-udef 98.7%

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

   9. times-frac 99.1%

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

6. Applied egg-rr 99.1%

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

   1. associate-*l/ 99.3%

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

   2. *-lft-identity 99.3%

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

8. Simplified 99.3%

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

9. Taylor expanded in v around 0 99.3%

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

10. Taylor expanded in t around 0 98.7%

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

    1. *-commutative 98.7%

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

    2. associate-*r* 98.6%

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

    3. associate-/r* 99.1%

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

    4. *-commutative 99.1%

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

12. Simplified 99.1%

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

13. Final simplification 99.1%

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

Alternative 5: 98.7% 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 99.1%

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

   1. associate-*r* 99.1%

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

   2. associate-*l* 99.1%

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

   3. cancel-sign-sub-inv 99.1%

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

   4. sqr-neg 99.1%

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

   5. metadata-eval 99.1%

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

   6. sqr-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 98.7%

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

   1. associate-*l* 98.7%

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

   2. unpow2 98.7%

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

   3. cancel-sign-sub-inv 98.7%

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

   4. metadata-eval 98.7%

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

   5. *-un-lft-identity 98.7%

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

   6. +-commutative 98.7%

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

   7. *-commutative 98.7%

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

   8. fma-udef 98.7%

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

   9. times-frac 99.1%

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

6. Applied egg-rr 99.1%

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

   1. associate-*l/ 99.3%

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

   2. *-lft-identity 99.3%

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

8. Simplified 99.3%

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

9. Taylor expanded in v around 0 99.3%

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

10. Final simplification 99.3%

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

Alternative 6: 97.8% accurate, 1.1× 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 99.1%

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

   1. associate-*l* 99.1%

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

   2. associate-*l* 99.1%

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

   3. associate-/r* 99.6%

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

   4. cancel-sign-sub-inv 99.6%

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

   5. sqr-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. *-commutative 99.6%

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

   8. fma-def 99.6%

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

   9. sqr-neg 99.6%

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

   10. metadata-eval 99.6%

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

   11. associate-*r* 99.6%

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

   12. *-commutative 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 98.1%

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

5. Final simplification 98.1%

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

Alternative 7: 97.8% accurate, 1.1× 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 99.1%

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

   1. associate-*l* 99.1%

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

   2. associate-*l* 99.1%

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

   3. associate-/r* 99.6%

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

   4. cancel-sign-sub-inv 99.6%

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

   5. sqr-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. *-commutative 99.6%

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

   8. fma-def 99.6%

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

   9. sqr-neg 99.6%

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

   10. metadata-eval 99.6%

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

   11. associate-*r* 99.6%

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

   12. *-commutative 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 98.1%

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

   1. associate-/r* 98.4%

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

   2. div-inv 98.2%

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

6. Applied egg-rr 98.2%

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

   1. un-div-inv 98.4%

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

8. Applied egg-rr 98.4%

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

9. Final simplification 98.4%

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

Reproduce

herbie shell --seed 2023336 
(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)))))