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

Percentage Accurate: 99.3% → 99.8%

Time: 8.5s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, t_] := N[(N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(1.0 / N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $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-*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-/r* 99.1%

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

   3. sub-neg 99.1%

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

   4. +-commutative 99.1%

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

   5. sqr-neg 99.1%

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

   6. *-commutative 99.1%

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

   7. distribute-rgt-neg-in 99.1%

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

   8. fma-def 99.1%

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

   9. sqr-neg 99.1%

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

   10. metadata-eval 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in t around 0 98.6%

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

   1. associate-*l/ 98.6%

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

   2. +-commutative 98.6%

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

   3. *-commutative 98.6%

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

   4. unpow2 98.6%

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

   5. associate-*r* 98.6%

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

   6. fma-udef 98.6%

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

   7. times-frac 98.9%

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

   8. +-commutative 98.9%

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

   9. *-commutative 98.9%

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

   10. unpow2 98.9%

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

   11. associate-*r* 98.9%

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

   12. fma-udef 98.9%

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

   13. unpow2 98.9%

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

6. Simplified 98.9%

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

   1. associate-*l/ 98.9%

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

   2. sqrt-div 99.9%

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

   3. metadata-eval 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.2%

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

4. Taylor expanded in t around 0 99.2%

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

   1. associate-*r* 99.2%

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

   2. unpow2 99.2%

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

   3. cancel-sign-sub-inv 99.2%

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

   4. metadata-eval 99.2%

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

   5. *-commutative 99.2%

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

   6. unpow2 99.2%

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

6. Simplified 99.2%

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

7. Final simplification 99.2%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 4: 98.4% 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%

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

3. Simplified 99.2%

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

4. Taylor expanded in v around 0 98.3%

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

5. Final simplification 98.3%

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

Alternative 5: 98.4% accurate, 1.1× speedup

Math

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

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(Float64(1.0 / 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[(N[(1.0 / t), $MachinePrecision] / N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

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

3. Simplified 99.2%

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

4. Taylor expanded in t around 0 99.2%

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

   1. associate-*r* 99.2%

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

   2. unpow2 99.2%

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

   3. cancel-sign-sub-inv 99.2%

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

   4. metadata-eval 99.2%

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

   5. *-commutative 99.2%

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

   6. unpow2 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in v around 0 98.3%

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

   1. associate-/r* 98.7%

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

9. Simplified 98.7%

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

10. Final simplification 98.7%

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

Alternative 6: 98.7% accurate, 1.1× speedup

Math

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

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(Float64(1.0 / 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[(N[(1.0 / Pi), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 99.1%

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

3. Simplified 99.2%

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

4. Taylor expanded in v around 0 98.3%

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

   1. inv-pow 98.3%

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

   2. *-commutative 98.3%

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

   3. unpow-prod-down 98.6%

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

   4. inv-pow 98.6%

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

   5. inv-pow 98.6%

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

6. Applied egg-rr 98.6%

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

   1. un-div-inv 99.0%

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

   2. associate-/r* 99.0%

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 7: 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-/r* 99.1%

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

   3. sub-neg 99.1%

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

   4. +-commutative 99.1%

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

   5. sqr-neg 99.1%

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

   6. *-commutative 99.1%

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

   7. distribute-rgt-neg-in 99.1%

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

   8. fma-def 99.1%

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

   9. sqr-neg 99.1%

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

   10. metadata-eval 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 97.8%

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

5. Final simplification 97.8%

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

Alternative 8: 97.9% 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-/r* 99.1%

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

   3. sub-neg 99.1%

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

   4. +-commutative 99.1%

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

   5. sqr-neg 99.1%

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

   6. *-commutative 99.1%

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

   7. distribute-rgt-neg-in 99.1%

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

   8. fma-def 99.1%

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

   9. sqr-neg 99.1%

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

   10. metadata-eval 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in t around 0 98.6%

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

   1. associate-*l/ 98.6%

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

   2. +-commutative 98.6%

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

   3. *-commutative 98.6%

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

   4. unpow2 98.6%

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

   5. associate-*r* 98.6%

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

   6. fma-udef 98.6%

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

   7. times-frac 98.9%

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

   8. +-commutative 98.9%

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

   9. *-commutative 98.9%

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

   10. unpow2 98.9%

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

   11. associate-*r* 98.9%

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

   12. fma-udef 98.9%

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

   13. unpow2 98.9%

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

6. Simplified 98.9%

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

   1. associate-*l/ 98.9%

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

   2. sqrt-div 99.9%

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

   3. metadata-eval 99.9%

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

8. Applied egg-rr 99.9%

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

9. Taylor expanded in v around 0 97.8%

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

    1. associate-/r* 98.0%

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

11. Simplified 98.0%

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

12. Final simplification 98.0%

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

Reproduce

herbie shell --seed 2023283 
(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)))))