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

Percentage Accurate: 99.3% → 99.8%

Time: 3.7s

Alternatives: 8

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

5. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-fma.f64 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

7. Applied rewrites 99.8%

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

Alternative 2: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

5. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-fma.f64 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.4%

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

Alternative 3: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, t_] := N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(N[(N[(Pi * t), $MachinePrecision] * N[Sqrt[N[(-6.0 * N[(v * v), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $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. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 99.3

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

4. Applied rewrites 99.3%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 99.3

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

6. Applied rewrites 99.3%

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

Alternative 4: 98.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 99.3

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in v around 0

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

7. Applied rewrites 98.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

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

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 98.2

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lift-*.f64 N/A

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

9. Applied rewrites 98.2%

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

    1. pow2 98.2

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

    2. *-commutative 98.2

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

    3. pow2 98.2

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

    4. pow2 98.2

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

    5. *-commutative 98.2

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

    6. pow2 98.2

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

    7. lift-*.f64 N/A

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

11. Applied rewrites 98.3%

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

Alternative 5: 98.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0

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

4. Applied rewrites 98.2%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 98.2

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 98.2

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 98.2

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

6. Applied rewrites 98.2%

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

7. Taylor expanded in v around 0

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

9. Applied rewrites 98.3%

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

Alternative 6: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, t_] := N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(N[(N[Sqrt[2.0], $MachinePrecision] * Pi), $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. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 98.3

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

4. Applied rewrites 98.3%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 98.3

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

6. Applied rewrites 98.3%

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

Alternative 7: 98.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(1.0 / N[(N[(N[Sqrt[2.0], $MachinePrecision] * Pi), $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. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in v around 0

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

   1. pow2 98.2

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

   2. metadata-eval 98.2

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

   3. fp-cancel-sign-sub-inv 98.2

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

   4. +-commutative 98.2

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

   5. pow2 98.2

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

7. Applied rewrites 98.2%

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

Alternative 8: 98.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(1.0 / N[(N[(t * Pi), $MachinePrecision] * N[Sqrt[2.0], $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. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in v around 0

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

   1. pow2 98.2

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

   2. metadata-eval 98.2

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

   3. fp-cancel-sign-sub-inv 98.2

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

   4. +-commutative 98.2

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

   5. pow2 98.2

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

7. Applied rewrites 98.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-sqrt.f64 98.1

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

9. Applied rewrites 98.1%

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

Reproduce

herbie shell --seed 2025101 
(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)))))