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

Percentage Accurate: 99.4% → 99.8%

Time: 4.5s

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: 99.4% 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 99.4%

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

   1. 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

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

FPCore

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

C

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

Julia

function code(v, t)
	return Float64(Float64(Float64(fma(Float64(v * v), 5.0, -1.0) / fma(v, v, -1.0)) / pi) / Float64(sqrt(Float64(fma(-3.0, Float64(v * v), 1.0) * 2.0)) * 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] / Pi), $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 99.4%

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

   1. 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. associate-*l* N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

3. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. 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. associate-/r* N/A

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

   4. frac-2neg N/A

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

   5. lift--.f64 N/A

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

   6. sub-negate-rev N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(v, t)
	return Float64(fma(Float64(v * v), 5.0, -1.0) / Float64(fma(v, v, -1.0) * Float64(Float64(pi * sqrt(fma(-6.0, Float64(v * v), 2.0))) * 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[(Pi * N[Sqrt[N[(-6.0 * N[(v * v), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. metadata-eval N/A

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

   10. fp-cancel-sub-sign-inv N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

5. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.5%

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. distribute-lft-in N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval 99.5%

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

7. Applied rewrites 99.5%

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

Alternative 5: 98.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. 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. associate-/r* N/A

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

   4. frac-2neg N/A

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

   5. lift--.f64 N/A

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

   6. sub-negate-rev N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in v around 0

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

6. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 98.9%

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

8. Applied rewrites 98.9%

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

Alternative 6: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. 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. associate-/r* N/A

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

   4. frac-2neg N/A

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

   5. lift--.f64 N/A

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

   6. sub-negate-rev N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in v around 0

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

6. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 98.9%

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

8. Applied rewrites 98.9%

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

9. Taylor expanded in v around 0

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

    1. lower-*.f64 98.9%

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

11. Applied rewrites 98.9%

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

Alternative 7: 98.9% 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 99.4%

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

2. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-sqrt.f64 98.5%

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

4. Applied rewrites 98.5%

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

5. Evaluated real constant 98.5%

   \[\leadsto \frac{1}{t \cdot 4.442882938158366} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{t \cdot \frac{1250560371546297}{281474976710656}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{t \cdot \color{blue}{\frac{1250560371546297}{281474976710656}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{\frac{1250560371546297}{281474976710656} \cdot \color{blue}{t}} \]

   4. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{\frac{1250560371546297}{281474976710656}}}{\color{blue}{t}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{\frac{1250560371546297}{281474976710656}}}{\color{blue}{t}} \]

   6. metadata-eval 98.9%

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

7. Applied rewrites 98.9%

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

Reproduce

herbie shell --seed 2025189 
(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)))))