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

Percentage Accurate: 99.3% → 99.5%

Time: 9.4s

Alternatives: 13

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: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* (* v v) 5.0))
  (* (* 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 - ((v * v) * 5.0)) / ((((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 - ((v * v) * 5.0)) / ((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 - ((v * v) * 5.0)) / ((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(Float64(v * v) * 5.0)) / Float64(Float64(pi * t) * Float64(Float64(1.0 - Float64(v * v)) * sqrt(Float64(2.0 * Float64(1.0 - Float64(Float64(v * v) * 3.0)))))))
end

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. associate-*l* 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

2. Final simplification 99.2%

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

Alternative 6: 98.2% 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 99.2%

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

   1. associate-*l* 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in v around 0 97.4%

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

5. Final simplification 97.4%

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

Alternative 7: 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 99.2%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 97.4%

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

5. Taylor expanded in v around 0 97.4%

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

   1. associate-/r* 97.8%

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

7. Simplified 97.8%

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

8. Final simplification 97.8%

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

Alternative 8: 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 99.2%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 97.1%

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

5. Final simplification 97.1%

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

Alternative 9: 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 99.2%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 97.1%

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

   1. div-inv 97.2%

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

   2. associate-/r* 97.1%

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

6. Applied egg-rr 97.1%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{\frac{1}{t}}{\pi}} \]
7. Step-by-step derivation

   1. associate-/r* 97.2%

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

8. Simplified 97.2%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{1}{t \cdot \pi}} \]
9. Step-by-step derivation

   1. un-div-inv 97.1%

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

   2. associate-/r* 97.2%

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

10. Applied egg-rr 97.2%

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

11. Final simplification 97.2%

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

Alternative 10: 20.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 97.3%

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

6. Applied egg-rr 20.4%

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

   1. *-commutative 20.4%

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

   2. distribute-rgt1-in 20.4%

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

   3. *-commutative 20.4%

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

8. Simplified 20.4%

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

9. Final simplification 20.4%

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

Alternative 11: 20.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(0.5 / t), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 97.4%

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

5. Taylor expanded in v around 0 97.4%

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

   1. associate-/r* 97.8%

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

7. Simplified 97.8%

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

   1. associate-/r* 97.4%

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

   2. frac-2neg 97.4%

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

   3. metadata-eval 97.4%

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

   4. div-inv 97.4%

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

   5. distribute-rgt-neg-in 97.4%

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

   6. distribute-rgt-neg-in 97.4%

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

9. Applied egg-rr 97.4%

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

    1. associate-*r/ 97.4%

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

    2. metadata-eval 97.4%

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

    3. associate-/r* 97.8%

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

    4. *-rgt-identity 97.8%

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

    5. times-frac 97.5%

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

    6. associate-/r* 97.6%

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

    7. associate-*l/ 97.7%

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

    8. associate-/r* 97.7%

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

    9. associate-*r/ 97.7%

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

    10. metadata-eval 97.7%

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

11. Simplified 97.7%

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

13. Applied egg-rr 20.3%

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

14. Final simplification 20.3%

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

Alternative 12: 17.0% accurate, 75.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{t} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ 1.0 t))

C

double code(double v, double t) {
	return 1.0 / t;
}

Fortran

real(8) function code(v, t)
    real(8), intent (in) :: v
    real(8), intent (in) :: t
    code = 1.0d0 / t
end function

Java

public static double code(double v, double t) {
	return 1.0 / t;
}

Python

def code(v, t):
	return 1.0 / t

Julia

function code(v, t)
	return Float64(1.0 / t)
end

MATLAB

function tmp = code(v, t)
	tmp = 1.0 / t;
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 97.4%

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

5. Taylor expanded in v around 0 97.4%

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

   1. associate-/r* 97.8%

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

7. Simplified 97.8%

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

   1. *-un-lft-identity 97.8%

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

   2. times-frac 97.7%

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

9. Applied egg-rr 97.7%

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

11. Applied egg-rr 17.0%

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

    1. associate-/r* 17.0%

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

    2. *-inverses 17.0%

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

13. Simplified 17.0%

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

14. Final simplification 17.0%

    \[\leadsto \frac{1}{t} \]

Alternative 13: 3.5% accurate, 226.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (v t) :precision binary64 1.0)

C

double code(double v, double t) {
	return 1.0;
}

Fortran

real(8) function code(v, t)
    real(8), intent (in) :: v
    real(8), intent (in) :: t
    code = 1.0d0
end function

Java

public static double code(double v, double t) {
	return 1.0;
}

Python

def code(v, t):
	return 1.0

Julia

function code(v, t)
	return 1.0
end

MATLAB

function tmp = code(v, t)
	tmp = 1.0;
end

Wolfram

code[v_, t_] := 1.0

Derivation

1. Initial program 99.2%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 97.4%

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

5. Taylor expanded in v around 0 97.4%

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

   1. associate-/r* 97.8%

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

7. Simplified 97.8%

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

   1. *-un-lft-identity 97.8%

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

   2. times-frac 97.7%

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

9. Applied egg-rr 97.7%

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

11. Applied egg-rr 3.7%

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

    1. *-inverses 3.7%

       \[\leadsto \color{blue}{1} \]

13. Simplified 3.7%

    \[\leadsto \color{blue}{1} \]

14. Final simplification 3.7%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023306 
(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)))))