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

Percentage Accurate: 99.3% → 99.5%

Time: 9.4s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

   1. associate-*l* 98.9%

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

   2. associate-/r* 99.1%

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

   3. sub-neg 99.1%

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

   4. +-commutative 99.1%

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

   5. sqr-neg 99.1%

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

   6. *-commutative 99.1%

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

   7. distribute-rgt-neg-in 99.1%

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

   8. fma-def 99.1%

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

   9. sqr-neg 99.1%

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

   10. metadata-eval 99.1%

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

3. Simplified 99.1%

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

   1. div-inv 99.1%

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

   2. fma-udef 99.1%

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

   3. associate-*l* 99.1%

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

   4. fma-def 99.1%

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

   5. *-commutative 99.1%

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

   6. +-commutative 99.1%

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

   7. associate-*l* 99.1%

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

   8. fma-def 99.1%

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

5. Applied egg-rr 99.1%

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

   1. associate-*l/ 99.1%

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

   2. associate-/r* 99.1%

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

7. Applied egg-rr 99.1%

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

   1. times-frac 99.5%

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

   2. associate-/l/ 99.5%

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

   3. *-commutative 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

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(Float64(v * 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[(N[(v * v), $MachinePrecision] * -6.0 + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.9%

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

   1. expm1-log1p-u 73.1%

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

   2. expm1-udef 29.5%

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

   3. cancel-sign-sub-inv 29.5%

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

   4. metadata-eval 29.5%

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

3. Applied egg-rr 29.5%

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

   1. expm1-def 73.1%

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

   2. expm1-log1p 98.9%

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

   3. *-commutative 98.9%

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

   4. associate-*l* 99.1%

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

   5. +-commutative 99.1%

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

   6. distribute-lft-in 99.1%

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

   7. unpow2 99.1%

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

   8. associate-*r* 99.1%

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

   9. metadata-eval 99.1%

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

   10. *-commutative 99.1%

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

   11. metadata-eval 99.1%

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

   12. fma-def 99.1%

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

   13. unpow2 99.1%

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Alternative 3: 99.3% accurate, 0.7× speedup

Math

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

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

Derivation

1. Initial program 98.9%

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

   1. associate-*l* 98.9%

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

   2. associate-/r* 99.1%

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

   3. sub-neg 99.1%

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

   4. +-commutative 99.1%

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

   5. sqr-neg 99.1%

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

   6. *-commutative 99.1%

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

   7. distribute-rgt-neg-in 99.1%

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

   8. fma-def 99.1%

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

   9. sqr-neg 99.1%

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

   10. metadata-eval 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double v, double t) {
	return 1.0 / ((Math.sqrt((2.0 + ((v * v) * -6.0))) * (t * (Math.PI * (1.0 - (v * v))))) / (1.0 + (-5.0 * (v * v))));
}

Python

def code(v, t):
	return 1.0 / ((math.sqrt((2.0 + ((v * v) * -6.0))) * (t * (math.pi * (1.0 - (v * v))))) / (1.0 + (-5.0 * (v * v))))

Julia

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

MATLAB

function tmp = code(v, t)
	tmp = 1.0 / ((sqrt((2.0 + ((v * v) * -6.0))) * (t * (pi * (1.0 - (v * v))))) / (1.0 + (-5.0 * (v * v))));
end

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Simplified 99.0%

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

5. Applied egg-rr 99.0%

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

   1. unpow-1 99.0%

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

   2. associate-/l* 98.9%

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

   3. associate-*l* 99.0%

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

   4. distribute-rgt-in 99.0%

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

   5. metadata-eval 99.0%

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

   6. unpow2 99.0%

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

   7. *-commutative 99.0%

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

   8. unpow2 99.0%

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

7. Simplified 99.0%

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

8. Taylor expanded in t around 0 98.9%

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

   1. associate-*l/ 98.9%

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

   2. unpow2 98.9%

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

   3. *-commutative 98.9%

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

   4. unpow2 98.9%

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

   5. *-commutative 98.9%

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

   6. unpow2 98.9%

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

10. Simplified 98.9%

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

11. Final simplification 98.9%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 6: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Simplified 99.0%

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

4. Taylor expanded in v around 0 98.0%

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

   1. *-commutative 98.0%

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

   2. associate-*l* 98.0%

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

   3. *-commutative 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in t around 0 98.0%

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

   1. associate-/r* 98.5%

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

9. Simplified 98.5%

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

    1. div-inv 98.5%

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

11. Applied egg-rr 98.5%

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

12. Final simplification 98.5%

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

Alternative 7: 98.3% 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 98.9%

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

3. Simplified 99.0%

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

4. Taylor expanded in v around 0 98.0%

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

5. Final simplification 98.0%

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

Alternative 8: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Simplified 99.0%

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

4. Taylor expanded in v around 0 98.0%

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

   1. *-commutative 98.0%

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

   2. associate-*l* 98.0%

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

   3. *-commutative 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in t around 0 98.0%

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

   1. associate-/r* 98.5%

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

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

Alternative 9: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

   1. associate-*l* 98.9%

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

   2. associate-/r* 99.1%

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

   3. sub-neg 99.1%

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

   4. +-commutative 99.1%

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

   5. sqr-neg 99.1%

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

   6. *-commutative 99.1%

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

   7. distribute-rgt-neg-in 99.1%

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

   8. fma-def 99.1%

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

   9. sqr-neg 99.1%

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

   10. metadata-eval 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in v around 0 97.6%

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

5. Final simplification 97.6%

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

Alternative 10: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

   1. expm1-log1p-u 73.1%

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

   2. expm1-udef 29.5%

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

   3. cancel-sign-sub-inv 29.5%

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

   4. metadata-eval 29.5%

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

3. Applied egg-rr 29.5%

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

   1. expm1-def 73.1%

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

   2. expm1-log1p 98.9%

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

   3. *-commutative 98.9%

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

   4. associate-*l* 99.1%

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

   5. +-commutative 99.1%

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

   6. distribute-lft-in 99.1%

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

   7. unpow2 99.1%

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

   8. associate-*r* 99.1%

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

   9. metadata-eval 99.1%

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

   10. *-commutative 99.1%

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

   11. metadata-eval 99.1%

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

   12. fma-def 99.1%

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

   13. unpow2 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in v around 0 97.6%

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

   1. associate-/r* 97.9%

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

8. Simplified 97.9%

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

9. Final simplification 97.9%

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

Reproduce

herbie shell --seed 2023297 
(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)))))