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

Percentage Accurate: 99.3% → 99.5%

Time: 4.6s

Alternatives: 7

Speedup: 0.4×

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: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 99.2%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\mathsf{fma}\left(\left(\left(\sqrt{2} \cdot \pi\right) \cdot t\right) \cdot -2.5, v \cdot v, \left(\sqrt{2} \cdot \pi\right) \cdot t\right)}} \]

6. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

8. Applied rewrites 99.3%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{fma}\left(-2.5 \cdot \left(v \cdot v\right), \sqrt{2} \cdot \pi, \sqrt{2} \cdot \pi\right) \cdot \color{blue}{t}} \]
9. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

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

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 99.3

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, -5, 1\right)}{\mathsf{fma}\left(-2.5 \cdot \left(v \cdot v\right), \sqrt{2} \cdot \pi, \sqrt{2} \cdot \pi\right) \cdot t} \]

10. Applied rewrites 99.3%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\mathsf{fma}\left(-2.5 \cdot \left(v \cdot v\right), \sqrt{2} \cdot \pi, \sqrt{2} \cdot \pi\right) \cdot t} \]

11. Taylor expanded in t around 0

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

    1. pow2 N/A

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

    2. *-commutative N/A

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

    3. pow2 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

13. Applied rewrites 99.6%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2} \cdot \pi\\ \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(-2.5 \cdot \left(v \cdot v\right), t\_1, t\_1\right) \cdot t} \end{array} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (let* ((t_1 (* (sqrt 2.0) PI)))
   (/ (fma (* v v) -5.0 1.0) (* (fma (* -2.5 (* v v)) t_1 t_1) t))))

C

double code(double v, double t) {
	double t_1 = sqrt(2.0) * ((double) M_PI);
	return fma((v * v), -5.0, 1.0) / (fma((-2.5 * (v * v)), t_1, t_1) * t);
}

Julia

function code(v, t)
	t_1 = Float64(sqrt(2.0) * pi)
	return Float64(fma(Float64(v * v), -5.0, 1.0) / Float64(fma(Float64(-2.5 * Float64(v * v)), t_1, t_1) * t))
end

Wolfram

code[v_, t_] := Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(N[(N[(-2.5 * N[(v * v), $MachinePrecision]), $MachinePrecision] * t$95$1 + t$95$1), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 99.2%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\mathsf{fma}\left(\left(\left(\sqrt{2} \cdot \pi\right) \cdot t\right) \cdot -2.5, v \cdot v, \left(\sqrt{2} \cdot \pi\right) \cdot t\right)}} \]

6. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

8. Applied rewrites 99.3%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{fma}\left(-2.5 \cdot \left(v \cdot v\right), \sqrt{2} \cdot \pi, \sqrt{2} \cdot \pi\right) \cdot \color{blue}{t}} \]
9. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

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

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 99.3

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, -5, 1\right)}{\mathsf{fma}\left(-2.5 \cdot \left(v \cdot v\right), \sqrt{2} \cdot \pi, \sqrt{2} \cdot \pi\right) \cdot t} \]

10. Applied rewrites 99.3%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\mathsf{fma}\left(-2.5 \cdot \left(v \cdot v\right), \sqrt{2} \cdot \pi, \sqrt{2} \cdot \pi\right) \cdot t} \]
11. Add Preprocessing

Alternative 3: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\left(\mathsf{fma}\left(v \cdot v, -2.5, 1\right) \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot t} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/ (fma -5.0 (* v v) 1.0) (* (* (fma (* v v) -2.5 1.0) (* (sqrt 2.0) PI)) t)))

C

double code(double v, double t) {
	return fma(-5.0, (v * v), 1.0) / ((fma((v * v), -2.5, 1.0) * (sqrt(2.0) * ((double) M_PI))) * t);
}

Julia

function code(v, t)
	return Float64(fma(-5.0, Float64(v * v), 1.0) / Float64(Float64(fma(Float64(v * v), -2.5, 1.0) * Float64(sqrt(2.0) * pi)) * t))
end

Wolfram

code[v_, t_] := N[(N[(-5.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(N[(v * v), $MachinePrecision] * -2.5 + 1.0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 99.2%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\mathsf{fma}\left(\left(\left(\sqrt{2} \cdot \pi\right) \cdot t\right) \cdot -2.5, v \cdot v, \left(\sqrt{2} \cdot \pi\right) \cdot t\right)}} \]

6. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

8. Applied rewrites 99.3%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{fma}\left(-2.5 \cdot \left(v \cdot v\right), \sqrt{2} \cdot \pi, \sqrt{2} \cdot \pi\right) \cdot \color{blue}{t}} \]
9. Step-by-step derivation

10. Applied rewrites 99.3%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\left(\mathsf{fma}\left(v \cdot v, -2.5, 1\right) \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot t}} \]
11. Add Preprocessing

Alternative 4: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. pow-flip N/A

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

   5. metadata-eval N/A

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

   6. lower-pow.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 51.5

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

5. Applied rewrites 51.5%

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

6. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 51.5

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

8. Applied rewrites 51.5%

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

9. Taylor expanded in v around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. pow2 N/A

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

    4. lift-*.f64 99.1

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

11. Applied rewrites 99.1%

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

Alternative 5: 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 99.1%

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

3. Taylor expanded in v around 0

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]
4. 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 99.1

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

5. Applied rewrites 99.1%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\sqrt{2} \cdot \pi\right) \cdot t}} \]
6. 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 99.1

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

7. Applied rewrites 99.1%

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

Alternative 6: 98.3% 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 99.1%

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

3. Taylor expanded in v around 0

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]
4. 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 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 99.1%

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

Alternative 7: 98.2% 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 99.1%

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

3. Taylor expanded in v around 0

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]
4. 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 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 99.1%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-*.f64 98.9

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

10. Applied rewrites 98.9%

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

Reproduce

herbie shell --seed 2025064 
(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)))))