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

Percentage Accurate: 99.3% → 99.4%

Time: 10.0s

Alternatives: 7

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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(v, t)
	return Float64(Float64(fma(Float64(v * v), -5.0, 1.0) / sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0)))) / Float64(pi * 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] / N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi * 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 99.5%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. expm1-log1p-u 99.3%

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

   2. expm1-undefine 99.3%

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

   3. log1p-undefine 99.3%

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

   4. add-exp-log 99.3%

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

   5. add-sqr-sqrt 44.3%

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

   6. sqrt-prod 99.0%

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

   7. sqr-neg 99.0%

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

   8. sqrt-unprod 54.8%

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

   9. add-sqr-sqrt 98.9%

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

   10. add-sqr-sqrt 54.8%

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

   11. sqrt-unprod 99.0%

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

   12. sqr-neg 99.0%

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

   13. sqrt-prod 44.3%

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

   14. add-sqr-sqrt 99.3%

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

   15. *-commutative 99.3%

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

6. Applied egg-rr 99.3%

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

7. Final simplification 99.3%

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

Alternative 3: 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 99.3%

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

3. Final simplification 99.3%

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

Alternative 4: 98.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in v around 0 98.5%

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

   1. add-cbrt-cube 99.1%

      \[\leadsto \frac{\sqrt{0.5}}{t \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}} \]

   2. pow3 99.1%

      \[\leadsto \frac{\sqrt{0.5}}{t \cdot \sqrt[3]{\color{blue}{{\pi}^{3}}}} \]

7. Applied egg-rr 99.1%

   \[\leadsto \frac{\sqrt{0.5}}{t \cdot \color{blue}{\sqrt[3]{{\pi}^{3}}}} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 98.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{0.5}} \cdot \sqrt{\sqrt{0.5}}}}{t \cdot \sqrt[3]{{\pi}^{3}}} \]

   2. rem-cbrt-cube 97.5%

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

   3. *-commutative 97.5%

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

   4. times-frac 98.1%

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

   5. pow1/2 98.1%

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

   6. sqrt-pow1 98.1%

      \[\leadsto \frac{\color{blue}{{0.5}^{\left(\frac{0.5}{2}\right)}}}{\pi} \cdot \frac{\sqrt{\sqrt{0.5}}}{t} \]

   7. metadata-eval 98.1%

      \[\leadsto \frac{{0.5}^{\color{blue}{0.25}}}{\pi} \cdot \frac{\sqrt{\sqrt{0.5}}}{t} \]

   8. pow1/2 98.1%

      \[\leadsto \frac{{0.5}^{0.25}}{\pi} \cdot \frac{\sqrt{\color{blue}{{0.5}^{0.5}}}}{t} \]

   9. sqrt-pow1 99.1%

      \[\leadsto \frac{{0.5}^{0.25}}{\pi} \cdot \frac{\color{blue}{{0.5}^{\left(\frac{0.5}{2}\right)}}}{t} \]

   10. metadata-eval 99.1%

       \[\leadsto \frac{{0.5}^{0.25}}{\pi} \cdot \frac{{0.5}^{\color{blue}{0.25}}}{t} \]

9. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{{0.5}^{0.25}}{\pi} \cdot \frac{{0.5}^{0.25}}{t}} \]
10. Step-by-step derivation

    1. frac-times 99.1%

       \[\leadsto \color{blue}{\frac{{0.5}^{0.25} \cdot {0.5}^{0.25}}{\pi \cdot t}} \]

    2. pow-prod-up 98.5%

       \[\leadsto \frac{\color{blue}{{0.5}^{\left(0.25 + 0.25\right)}}}{\pi \cdot t} \]

    3. metadata-eval 98.5%

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

    4. pow1/2 98.5%

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

    5. *-rgt-identity 98.5%

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

    6. frac-times 98.5%

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

    7. clear-num 99.2%

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

    8. associate-*l/ 99.1%

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

    9. *-un-lft-identity 99.1%

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

11. Applied egg-rr 99.1%

    \[\leadsto \color{blue}{\frac{\frac{1}{t}}{\frac{\pi}{\sqrt{0.5}}}} \]
12. Add Preprocessing

Alternative 5: 98.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(1.0 / N[(t * N[(Pi / N[Sqrt[0.5], $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 99.4%

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

5. Taylor expanded in v around 0 98.5%

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

   1. clear-num 98.6%

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

   2. inv-pow 98.6%

      \[\leadsto \color{blue}{{\left(\frac{t \cdot \pi}{\sqrt{0.5}}\right)}^{-1}} \]

7. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{{\left(\frac{t \cdot \pi}{\sqrt{0.5}}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 98.6%

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

   2. associate-/l* 99.1%

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

9. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{1}{t \cdot \frac{\pi}{\sqrt{0.5}}}} \]
10. Add Preprocessing

Alternative 6: 97.9% accurate, 1.2× 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 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 98.5%

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

   1. div-inv 98.5%

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

7. Applied egg-rr 98.5%

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

   1. div-inv 98.5%

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

   2. associate-/r* 98.6%

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

9. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{0.5}}{t}}{\pi}} \]
10. Add Preprocessing

Alternative 7: 97.9% accurate, 1.2× 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 99.4%

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

5. Taylor expanded in v around 0 98.5%

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

6. Final simplification 98.5%

   \[\leadsto \frac{\sqrt{0.5}}{\pi \cdot t} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(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)))))