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

Percentage Accurate: 99.3% → 99.8%

Time: 4.4s

Alternatives: 7

Speedup: 1.1×

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, t_] := N[(N[(N[(-5.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(-3.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * t), $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. frac-2neg N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. associate-*r* N/A

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

3. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, t_] := N[(N[(N[(-5.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(-3.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * Pi), $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. frac-2neg N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

3. Applied rewrites 99.3%

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

Alternative 4: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 99.5%

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.5%

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

5. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-*.f64 N/A

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

7. Applied rewrites 99.5%

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

Alternative 5: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 99.5%

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.5%

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

5. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 99.3%

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.3%

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

7. Applied rewrites 99.3%

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

Alternative 6: 98.8% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (v t)
  :precision binary64
  (/ (/ (/ (fma (* v v) 5.0 -1.0) (fma v v -1.0)) (* (sqrt 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(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(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[2.0], $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. Taylor expanded in v around 0

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

6. Applied rewrites 98.8%

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

Alternative 7: 98.7% accurate, 8.7× 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

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

   \[\leadsto \frac{0.22507907903927651}{t} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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)))))