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

Percentage Accurate: 99.3% → 99.3%

Time: 8.6s

Alternatives: 6

Speedup: N/A×

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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. expm1-log1p-u 99.1%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(v \cdot v\right)\right)\right)}\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

   2. expm1-udef 99.1%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \left(v \cdot v\right)\right)} - 1\right)}\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

   3. log1p-udef 99.1%

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

   4. add-exp-log 99.1%

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

   5. *-commutative 99.1%

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

   6. associate-*r* 99.1%

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

3. Applied egg-rr 99.1%

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

   1. div-inv 99.1%

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

   2. cancel-sign-sub-inv 99.1%

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

   3. metadata-eval 99.1%

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

   4. *-commutative 99.1%

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

   5. associate-*l* 99.1%

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

   6. *-commutative 99.1%

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

   7. associate-*l* 99.1%

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

   8. associate--r- 99.1%

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

5. Applied egg-rr 99.1%

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

6. Final simplification 99.1%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 3: 98.3% accurate, 1.1× 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 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 98.4%

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

5. Final simplification 98.4%

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

Alternative 4: 98.4% accurate, 1.1× speedup

Math

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

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(Float64(1.0 / 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[(N[(1.0 / t), $MachinePrecision] / N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 98.4%

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

   1. expm1-log1p-u 71.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\right)\right)} \]

   2. expm1-udef 31.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\right)} - 1} \]

6. Applied egg-rr 31.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 71.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\right)\right)} \]

   2. expm1-log1p 98.4%

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

   3. associate-/r* 98.7%

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

8. Simplified 98.7%

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

9. Final simplification 98.7%

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

Alternative 5: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{0.5}}{\pi \cdot t} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ (sqrt 0.5) (* PI t)))

C

double code(double v, double t) {
	return sqrt(0.5) / (((double) M_PI) * t);
}

Java

public static double code(double v, double t) {
	return Math.sqrt(0.5) / (Math.PI * t);
}

Python

def code(v, t):
	return math.sqrt(0.5) / (math.pi * t)

Julia

function code(v, t)
	return Float64(sqrt(0.5) / Float64(pi * t))
end

MATLAB

function tmp = code(v, t)
	tmp = sqrt(0.5) / (pi * t);
end

Wolfram

code[v_, t_] := N[(N[Sqrt[0.5], $MachinePrecision] / N[(Pi * 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. Step-by-step derivation

   1. associate-*l* 99.1%

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

   2. associate-/r* 99.1%

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

   3. sub-neg 99.1%

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

   4. +-commutative 99.1%

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

   5. sqr-neg 99.1%

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

   6. *-commutative 99.1%

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

   7. distribute-rgt-neg-in 99.1%

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

   8. fma-def 99.1%

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

   9. sqr-neg 99.1%

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

   10. metadata-eval 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 97.9%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]

5. Final simplification 97.9%

   \[\leadsto \frac{\sqrt{0.5}}{\pi \cdot t} \]

Alternative 6: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\sqrt{0.5}}{t}}{\pi} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ (/ (sqrt 0.5) t) PI))

C

double code(double v, double t) {
	return (sqrt(0.5) / t) / ((double) M_PI);
}

Java

public static double code(double v, double t) {
	return (Math.sqrt(0.5) / t) / Math.PI;
}

Python

def code(v, t):
	return (math.sqrt(0.5) / t) / math.pi

Julia

function code(v, t)
	return Float64(Float64(sqrt(0.5) / t) / pi)
end

MATLAB

function tmp = code(v, t)
	tmp = (sqrt(0.5) / t) / pi;
end

Wolfram

code[v_, t_] := N[(N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision] / Pi), $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. Step-by-step derivation

   1. associate-*l* 99.1%

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

   2. associate-/r* 99.1%

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

   3. sub-neg 99.1%

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

   4. +-commutative 99.1%

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

   5. sqr-neg 99.1%

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

   6. *-commutative 99.1%

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

   7. distribute-rgt-neg-in 99.1%

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

   8. fma-def 99.1%

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

   9. sqr-neg 99.1%

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

   10. metadata-eval 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 97.9%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]
5. Step-by-step derivation

   1. div-inv 97.9%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{1}{t \cdot \pi}} \]

   2. associate-/r* 98.1%

      \[\leadsto \sqrt{0.5} \cdot \color{blue}{\frac{\frac{1}{t}}{\pi}} \]

6. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{\frac{1}{t}}{\pi}} \]

7. Taylor expanded in t around 0 97.9%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]
8. Step-by-step derivation

   1. associate-/r* 98.2%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{0.5}}{t}}{\pi}} \]

9. Simplified 98.2%

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{0.5}}{t}}{\pi}} \]

10. Final simplification 98.2%

    \[\leadsto \frac{\frac{\sqrt{0.5}}{t}}{\pi} \]

Reproduce

herbie shell --seed 2023271 
(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)))))