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

Percentage Accurate: 99.3% → 99.8%

Time: 3.4s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

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

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

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

3. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

3. Applied rewrites 99.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. lift--.f64 N/A

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

   7. sub-flip N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. remove-double-neg N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

3. Applied rewrites 99.5%

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

Alternative 5: 98.7% accurate, 9.3× speedup

Math

\[\frac{0.22507907903927651}{t} \]

FPCore

(FPCore (v t) :precision binary64 (/ 0.22507907903927651 t))

C

double code(double v, double t) {
	return 0.22507907903927651 / t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, t)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: t
    code = 0.22507907903927651d0 / t
end function

Java

public static double code(double v, double t) {
	return 0.22507907903927651 / t;
}

Python

def code(v, t):
	return 0.22507907903927651 / t

Julia

function code(v, t)
	return Float64(0.22507907903927651 / t)
end

MATLAB

function tmp = code(v, t)
	tmp = 0.22507907903927651 / t;
end

Wolfram

code[v_, t_] := N[(0.22507907903927651 / t), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-sqrt.f64 98.7

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

6. Applied rewrites 98.7%

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

7. Evaluated real constant 98.7%

   \[\leadsto \frac{\frac{8109328451921441}{36028797018963968}}{t} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025173 
(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)))))