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

Percentage Accurate: 99.3% → 99.6%

Time: 8.7s

Alternatives: 11

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: 99.6% 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 99.3%

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

3. 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 99.3%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

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 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 4: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{t} \cdot \frac{\sqrt{0.5}}{\pi} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (* (/ 1.0 t) (/ (sqrt 0.5) PI)))

C

double code(double v, double t) {
	return (1.0 / t) * (sqrt(0.5) / ((double) M_PI));
}

Java

public static double code(double v, double t) {
	return (1.0 / t) * (Math.sqrt(0.5) / Math.PI);
}

Python

def code(v, t):
	return (1.0 / t) * (math.sqrt(0.5) / math.pi)

Julia

function code(v, t)
	return Float64(Float64(1.0 / t) * Float64(sqrt(0.5) / pi))
end

MATLAB

function tmp = code(v, t)
	tmp = (1.0 / t) * (sqrt(0.5) / pi);
end

Wolfram

code[v_, t_] := N[(N[(1.0 / t), $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in v around 0 98.1%

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

   1. *-un-lft-identity 98.1%

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

   2. times-frac 98.2%

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

6. Applied egg-rr 98.2%

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

7. Final simplification 98.2%

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

Alternative 5: 98.4% 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 99.3%

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

   1. associate-*l* 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in v around 0 98.6%

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

   1. *-commutative 98.6%

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

6. Simplified 98.6%

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

7. Final simplification 98.6%

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

Alternative 6: 98.8% accurate, 1.1× speedup

Math

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

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(Float64(Float64(1.0 / 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[(N[(N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Pi), $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. Step-by-step derivation

   1. associate-*l* 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in v around 0 98.6%

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

   1. *-commutative 98.6%

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

6. Simplified 98.6%

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

7. Taylor expanded in t around 0 98.6%

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

   1. *-commutative 98.6%

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

   2. *-commutative 98.6%

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

   3. associate-/r* 99.0%

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

   4. associate-/r* 99.0%

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

9. Simplified 99.0%

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

10. Final simplification 99.0%

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

Alternative 7: 97.8% 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 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in v around 0 98.1%

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

5. Final simplification 98.1%

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

Alternative 8: 20.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-*l* 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in v around 0 98.6%

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

   1. *-commutative 98.6%

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

6. Simplified 98.6%

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

7. Taylor expanded in t around 0 98.6%

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

   1. *-commutative 98.6%

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

   2. *-commutative 98.6%

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

   3. associate-/r* 99.0%

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

   4. associate-/r* 99.0%

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

9. Simplified 99.0%

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

11. Applied egg-rr 20.3%

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

12. Final simplification 20.3%

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

Alternative 9: 20.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1.0 / Pi), $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. Step-by-step derivation

   1. associate-*l* 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in v around 0 98.6%

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

   1. *-commutative 98.6%

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

6. Simplified 98.6%

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

7. Taylor expanded in t around 0 98.6%

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

   1. *-commutative 98.6%

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

   2. *-commutative 98.6%

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

   3. associate-/r* 99.0%

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

   4. associate-/r* 99.0%

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

9. Simplified 99.0%

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

11. Applied egg-rr 20.4%

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

12. Final simplification 20.4%

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

Alternative 10: 15.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(Pi / 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. Step-by-step derivation

   1. associate-*l* 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in v around 0 98.6%

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

   1. *-commutative 98.6%

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

6. Simplified 98.6%

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

7. Taylor expanded in t around 0 98.6%

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

   1. *-commutative 98.6%

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

   2. *-commutative 98.6%

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

   3. associate-/r* 99.0%

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

   4. associate-/r* 99.0%

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

9. Simplified 99.0%

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

11. Applied egg-rr 15.7%

    \[\leadsto \frac{\color{blue}{\log \left(e^{\pi}\right)}}{t} \]
12. Step-by-step derivation

    1. rem-log-exp 15.7%

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

13. Simplified 15.7%

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

14. Final simplification 15.7%

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

Alternative 11: 3.8% accurate, 45.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in v around 0 98.6%

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

   1. *-commutative 98.6%

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

6. Simplified 98.6%

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

7. Taylor expanded in t around 0 98.6%

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

   1. associate-*r* 98.5%

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

   2. *-commutative 98.5%

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

   3. associate-*r* 98.5%

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

9. Simplified 98.5%

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

11. Applied egg-rr 3.7%

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

    1. *-rgt-identity 3.7%

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

    2. 1-exp 3.7%

       \[\leadsto \frac{1}{\frac{\pi \cdot \color{blue}{e^{0}}}{\pi \cdot t}} \]

    3. associate-*l/ 3.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{\pi}{\pi \cdot t} \cdot e^{0}}} \]

    4. 1-exp 3.7%

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

    5. *-inverses 3.7%

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

    6. times-frac 3.5%

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

    7. associate-/r* 3.7%

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

    8. *-inverses 3.7%

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

13. Simplified 3.7%

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

14. Final simplification 3.7%

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

Reproduce

herbie shell --seed 2023308 
(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)))))