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

Percentage Accurate: 99.3% → 99.4%

Time: 2.6s

Alternatives: 5

Speedup: 1.0×

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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v, t)
	tmp = (1.0 - (5.0 * (v * v))) / (((t * (pi * sqrt(2.0))) * sqrt((1.0 - (3.0 * (v ^ 2.0))))) * (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[(t * N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(3.0 * N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $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 t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 99.4

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

4. Applied rewrites 99.4%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 99.3

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

7. Applied rewrites 99.3%

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

Alternative 3: 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]

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. Add Preprocessing

Alternative 4: 98.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in v around 0

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 98.8

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

10. Applied rewrites 98.8%

    \[\leadsto \frac{\frac{1}{\pi \cdot \sqrt{2}}}{t} \]
11. Add Preprocessing

Alternative 5: 98.4% accurate, 3.4× speedup

Math

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

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(t * Float64(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[(t * N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]
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. lift-PI.f64 N/A

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

   5. lower-sqrt.f64 98.4

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

4. Applied rewrites 98.4%

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

Reproduce

herbie shell --seed 2025132 
(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)))))