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

Percentage Accurate: 99.3% → 99.3%

Time: 10.2s

Alternatives: 8

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: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Simplified 99.2%

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

Alternative 2: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

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 - (v * (v * t))));
}

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 - Float64(v * Float64(v * t)))))
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 * N[(v * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

3. Simplified 99.1%

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

5. Taylor expanded in v around 0 99.1%

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

   1. add-sqr-sqrt 46.4%

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

   2. fma-define 46.4%

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

   3. mul-1-neg 46.4%

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

   4. pow2 46.4%

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

   5. add-sqr-sqrt 46.4%

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

   6. sqrt-unprod 44.4%

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

   7. sqr-neg 44.4%

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

   8. pow2 44.4%

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

   9. mul-1-neg 44.4%

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

   10. pow2 44.4%

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

   11. mul-1-neg 44.4%

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

   12. sqrt-unprod 25.1%

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

   13. add-sqr-sqrt 45.7%

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

   14. *-commutative 45.7%

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

   15. pow2 45.7%

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

7. Applied egg-rr 99.1%

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

   1. *-un-lft-identity 99.1%

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

   2. *-commutative 99.1%

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

   3. pow2 99.1%

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

   4. *-commutative 99.1%

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

   5. associate-*l* 99.1%

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

   6. prod-diff 99.1%

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

9. Applied egg-rr 99.1%

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

    1. +-commutative 99.1%

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

    2. fma-undefine 99.1%

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

    3. distribute-lft-neg-in 99.1%

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

    4. neg-mul-1 99.1%

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

    5. associate-*r* 99.1%

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

    6. associate-*r* 99.1%

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

    7. distribute-lft1-in 99.1%

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

    8. metadata-eval 99.1%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\pi \cdot \left(\color{blue}{0} \cdot \left(v \cdot \left(t \cdot v\right)\right) + \mathsf{fma}\left(t, 1, -\left(v \cdot t\right) \cdot v\right)\right)} \]

    9. fma-undefine 99.1%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\pi \cdot \left(0 \cdot \left(v \cdot \left(t \cdot v\right)\right) + \color{blue}{\left(t \cdot 1 + \left(-\left(v \cdot t\right) \cdot v\right)\right)}\right)} \]

    10. associate-*r* 99.1%

        \[\leadsto \frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\pi \cdot \left(0 \cdot \left(v \cdot \left(t \cdot v\right)\right) + \left(t \cdot 1 + \left(-\color{blue}{v \cdot \left(t \cdot v\right)}\right)\right)\right)} \]

    11. *-rgt-identity 99.1%

        \[\leadsto \frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\pi \cdot \left(0 \cdot \left(v \cdot \left(t \cdot v\right)\right) + \left(\color{blue}{t} + \left(-v \cdot \left(t \cdot v\right)\right)\right)\right)} \]

    12. unsub-neg 99.1%

        \[\leadsto \frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\pi \cdot \left(0 \cdot \left(v \cdot \left(t \cdot v\right)\right) + \color{blue}{\left(t - v \cdot \left(t \cdot v\right)\right)}\right)} \]

11. Simplified 99.1%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\pi \cdot \color{blue}{\left(0 \cdot \left(v \cdot \left(t \cdot v\right)\right) + \left(t - v \cdot \left(t \cdot v\right)\right)\right)}} \]
12. Step-by-step derivation

    1. associate-+r- 99.1%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\pi \cdot \color{blue}{\left(\left(0 \cdot \left(v \cdot \left(t \cdot v\right)\right) + t\right) - v \cdot \left(t \cdot v\right)\right)}} \]

    2. mul0-lft 99.1%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\pi \cdot \left(\left(\color{blue}{0} + t\right) - v \cdot \left(t \cdot v\right)\right)} \]

    3. *-commutative 99.1%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\pi \cdot \left(\left(0 + t\right) - v \cdot \color{blue}{\left(v \cdot t\right)}\right)} \]

13. Applied egg-rr 99.1%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\pi \cdot \color{blue}{\left(\left(0 + t\right) - v \cdot \left(v \cdot t\right)\right)}} \]

14. Final simplification 99.1%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* v (* 5.0 v)))
  (* (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 - (v * (5.0 * v))) / (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 - (v * (5.0 * v))) / (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 - (v * (5.0 * v))) / (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(v * Float64(5.0 * v))) / 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 - (v * (5.0 * v))) / (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[(v * N[(5.0 * v), $MachinePrecision]), $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 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (* (- 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 - (5.0 * (v * v))) / ((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 - (5.0 * (v * v))) / ((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 - (5.0 * (v * v))) / ((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(5.0 * Float64(v * v))) / 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 - (5.0 * (v * v))) / ((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[(5.0 * N[(v * v), $MachinePrecision]), $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 98.8%

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

3. Final simplification 98.8%

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

Alternative 5: 98.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Simplified 99.2%

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

5. Taylor expanded in v around 0 97.1%

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

   1. *-un-lft-identity 97.1%

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

   2. times-frac 97.4%

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

7. Applied egg-rr 97.4%

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

   1. associate-*r/ 97.4%

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

   2. associate-*l/ 97.3%

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

   3. /-rgt-identity 97.3%

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

   4. clear-num 96.9%

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

   5. clear-num 96.9%

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

   6. frac-times 96.6%

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

   7. metadata-eval 96.6%

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

   8. div-inv 96.5%

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

   9. clear-num 96.6%

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

   10. /-rgt-identity 96.6%

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

   11. pow1/2 96.6%

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

   12. pow-flip 97.3%

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

   13. metadata-eval 97.3%

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

9. Applied egg-rr 97.3%

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

10. Taylor expanded in t around 0 97.4%

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

    1. *-commutative 97.4%

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

    2. associate-*l* 97.2%

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

    3. *-commutative 97.2%

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

    4. associate-/r* 98.1%

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

12. Simplified 98.1%

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

Alternative 6: 98.3% 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 98.8%

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

3. Simplified 99.2%

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

5. Taylor expanded in v around 0 97.1%

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

   1. *-un-lft-identity 97.1%

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

   2. times-frac 97.4%

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

7. Applied egg-rr 97.4%

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

   1. clear-num 98.0%

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

   2. un-div-inv 97.9%

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

9. Applied egg-rr 97.9%

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

Alternative 7: 97.8% 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 98.8%

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

3. Simplified 99.2%

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

5. Taylor expanded in v around 0 97.1%

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

   1. associate-/r* 97.4%

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

7. Simplified 97.4%

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

Alternative 8: 97.7% 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 98.8%

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

3. Simplified 99.2%

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

5. Taylor expanded in v around 0 97.1%

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

6. Final simplification 97.1%

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

Reproduce

herbie shell --seed 2024130 
(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)))))