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

Percentage Accurate: 99.3% → 99.6%

Time: 7.0s

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: 99.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (/ (fma v (* v -5.0) 1.0) (* PI (- 1.0 (* v v))))
  (* t (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) * (1.0 - (v * v)))) / (t * 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(1.0 - Float64(v * v)))) / Float64(t * 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[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Sqrt[N[(N[(v * v), $MachinePrecision] * -6.0 + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. associate-*r* 99.4%

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

   4. associate-/r* 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. expm1-log1p-u 73.8%

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

   2. expm1-udef 28.9%

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

   3. cancel-sign-sub-inv 28.9%

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

   4. metadata-eval 28.9%

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

3. Applied egg-rr 28.9%

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

   1. expm1-def 73.8%

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

   2. expm1-log1p 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*l* 99.5%

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

   5. +-commutative 99.5%

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

   6. distribute-rgt-in 99.5%

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

   7. unpow2 99.5%

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

   8. *-commutative 99.5%

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

   9. associate-*l* 99.5%

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

   10. unpow2 99.5%

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

   11. metadata-eval 99.5%

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

   12. metadata-eval 99.5%

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

   13. associate-*r* 99.5%

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

   14. fma-def 99.5%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.4%

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

   2. associate-/r* 99.3%

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

   3. associate-/l/ 99.3%

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

   4. sub-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. *-commutative 99.3%

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

   7. distribute-rgt-neg-in 99.3%

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

   8. fma-def 99.3%

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

   9. metadata-eval 99.3%

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

   10. sub-neg 99.3%

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

   11. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

   1. *-un-lft-identity 99.3%

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

   2. associate-/r* 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

2. Final simplification 99.4%

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

Alternative 5: 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 99.4%

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

2. Taylor expanded in v around 0 98.7%

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

3. Taylor expanded in v around 0 98.7%

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

   1. *-commutative 98.7%

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

   2. associate-*l* 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 6: 98.3% accurate, 1.1× 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 99.4%

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

2. Taylor expanded in v around 0 98.7%

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

3. Taylor expanded in v around 0 98.7%

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

   1. *-commutative 98.7%

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

   2. associate-*l* 98.8%

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

5. Simplified 98.8%

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

   1. inv-pow 98.8%

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

   2. associate-*r* 98.7%

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

   3. *-commutative 98.7%

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

   4. associate-*r* 98.7%

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

   5. unpow-prod-down 98.7%

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

   6. inv-pow 98.7%

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

   7. *-commutative 98.7%

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

   8. inv-pow 98.7%

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

7. Applied egg-rr 98.7%

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

   1. associate-*l/ 98.9%

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

   2. *-un-lft-identity 98.9%

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

9. Applied egg-rr 98.9%

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

10. Final simplification 98.9%

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

Alternative 7: 97.6% 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 99.4%

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

   1. associate-*l* 99.4%

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

   2. associate-/r* 99.3%

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

   3. associate-/l/ 99.3%

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

   4. sub-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. *-commutative 99.3%

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

   7. distribute-rgt-neg-in 99.3%

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

   8. fma-def 99.3%

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

   9. metadata-eval 99.3%

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

   10. sub-neg 99.3%

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

   11. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in v around 0 98.3%

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

5. Final simplification 98.3%

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

Reproduce

herbie shell --seed 2023217 
(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)))))