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

Percentage Accurate: 99.4% → 99.5%

Time: 22.2s

Alternatives: 9

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: 99.4% 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

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

FPCore

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

C

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

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))) * Float64(1.0 / sqrt(fma(Float64(v * v), -6.0, 2.0))))
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[(1.0 / 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. 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* 99.4%

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

   3. associate-/l/ 99.4%

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

   4. sub-neg 99.4%

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

   5. +-commutative 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-neg-in 99.4%

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

   8. fma-def 99.4%

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

   9. metadata-eval 99.4%

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

   10. sub-neg 99.4%

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

   11. distribute-rgt-in 99.4%

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

3. Simplified 99.4%

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

   1. div-inv 99.3%

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

   2. associate-/r* 99.5%

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

   3. +-commutative 99.5%

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

   4. fma-udef 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, t_] := N[(N[(N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $MachinePrecision] / Pi), $MachinePrecision] / N[(N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * N[(t * N[(1.0 - N[(v * v), $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. 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* 99.4%

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

   3. associate-/l/ 99.4%

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

   4. sub-neg 99.4%

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

   5. +-commutative 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-neg-in 99.4%

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

   8. fma-def 99.4%

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

   9. metadata-eval 99.4%

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

   10. sub-neg 99.4%

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

   11. distribute-rgt-in 99.4%

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

3. Simplified 99.4%

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

   1. expm1-log1p-u 73.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}\right)\right)} \]

   2. expm1-udef 28.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}\right)} - 1} \]

   3. associate-/l/ 28.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(1 - v \cdot v\right)}}\right)} - 1 \]

   4. associate-/r* 28.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi}}{t}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(1 - v \cdot v\right)}\right)} - 1 \]

   5. +-commutative 28.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi}}{t}}{\sqrt{\color{blue}{\left(v \cdot v\right) \cdot -6 + 2}} \cdot \left(1 - v \cdot v\right)}\right)} - 1 \]

   6. fma-udef 28.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi}}{t}}{\sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \cdot \left(1 - v \cdot v\right)}\right)} - 1 \]

5. Applied egg-rr 28.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi}}{t}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot \left(1 - v \cdot v\right)}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 73.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi}}{t}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot \left(1 - v \cdot v\right)}\right)\right)} \]

   2. expm1-log1p 99.6%

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

   3. associate-/l/ 99.5%

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

   4. fma-udef 99.5%

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

   5. associate-*r* 99.5%

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

   6. fma-udef 99.5%

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

   7. associate-*l* 99.5%

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

   8. fma-udef 99.5%

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

   9. associate-*r* 99.5%

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

   10. fma-def 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double v, double t) {
	return (1.0 - ((1.0 + ((v * v) * 5.0)) + -1.0)) / ((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(1.0 + Float64(Float64(v * v) * 5.0)) + -1.0)) / 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[(1.0 + N[(N[(v * v), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] + -1.0), $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 99.4%

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

   2. expm1-udef 99.4%

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

   3. log1p-udef 99.4%

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

   4. add-exp-log 99.4%

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

   5. *-commutative 99.4%

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

3. Applied egg-rr 99.4%

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

   1. expm1-log1p-u 69.7%

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

   2. expm1-udef 23.4%

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

   3. cancel-sign-sub-inv 23.4%

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

   4. metadata-eval 23.4%

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

5. Applied egg-rr 23.4%

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

   1. expm1-def 69.7%

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

   2. expm1-log1p 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*l* 99.4%

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

   5. +-commutative 99.4%

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

   6. distribute-rgt-in 99.4%

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

   7. unpow2 99.4%

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

   8. *-commutative 99.4%

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

   9. associate-*l* 99.4%

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

   10. unpow2 99.4%

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

   11. metadata-eval 99.4%

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

   12. metadata-eval 99.4%

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

   13. associate-*r* 99.4%

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

   14. fma-def 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1.0 + N[(v * N[(v * -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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]), $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 99.4%

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

   2. expm1-udef 99.4%

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

   3. log1p-udef 99.4%

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

   4. add-exp-log 99.4%

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

   5. *-commutative 99.4%

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

3. Applied egg-rr 99.4%

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

   1. div-sub 99.4%

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

   2. *-commutative 99.4%

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

   3. *-commutative 99.4%

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

   4. cancel-sign-sub-inv 99.4%

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

   5. metadata-eval 99.4%

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

   6. *-commutative 99.4%

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

5. Applied egg-rr 99.4%

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

   1. div-sub 99.4%

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

   2. sub-neg 99.4%

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

   3. distribute-rgt-neg-in 99.4%

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

   4. distribute-rgt-neg-in 99.4%

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

   5. metadata-eval 99.4%

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

   6. associate-*l* 99.4%

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

   7. *-commutative 99.4%

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

   8. associate-*r* 99.4%

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

   9. unpow2 99.4%

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

   10. *-commutative 99.4%

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

   11. *-commutative 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 5: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1.0 + N[(N[(v * v), $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(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] * N[(Pi * t), $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 99.4%

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

   2. expm1-udef 99.4%

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

   3. log1p-udef 99.4%

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

   4. add-exp-log 99.4%

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

   5. *-commutative 99.4%

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

3. Applied egg-rr 99.4%

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

   1. div-sub 99.4%

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

   2. *-commutative 99.4%

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

   3. *-commutative 99.4%

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

   4. cancel-sign-sub-inv 99.4%

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

   5. metadata-eval 99.4%

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

   6. *-commutative 99.4%

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

5. Applied egg-rr 99.4%

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

   1. div-sub 99.4%

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

   2. associate-*r* 99.4%

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

   3. unpow2 99.4%

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

   4. cancel-sign-sub-inv 99.4%

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

   5. distribute-lft-neg-in 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. unpow2 99.4%

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

   8. metadata-eval 99.4%

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

   9. *-commutative 99.4%

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

   10. *-commutative 99.4%

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

   11. associate-*r* 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 6: 98.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1.0 / N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / 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. Taylor expanded in v around 0 97.9%

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

   1. *-commutative 97.9%

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

   2. *-commutative 97.9%

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

   3. associate-*l* 97.9%

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

   4. *-commutative 97.9%

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

4. Simplified 97.9%

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

   1. inv-pow 97.9%

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

   2. associate-*r* 98.0%

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

   3. unpow-prod-down 98.1%

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

   4. inv-pow 98.1%

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

   5. inv-pow 98.1%

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

6. Applied egg-rr 98.1%

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

7. Final simplification 98.1%

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

Alternative 7: 98.6% 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 98.0%

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

   1. *-commutative 98.0%

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

   2. *-commutative 98.0%

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

   3. associate-*l* 98.0%

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

   4. *-commutative 98.0%

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

4. Simplified 98.0%

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

5. Taylor expanded in v around 0 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l* 98.0%

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

7. Simplified 98.0%

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

8. Final simplification 98.0%

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

Alternative 8: 98.6% 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 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 98.0%

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

   1. *-commutative 98.0%

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

   2. *-commutative 98.0%

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

   3. associate-*l* 98.0%

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

   4. *-commutative 98.0%

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

4. Simplified 98.0%

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

5. Taylor expanded in v around 0 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l* 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in t around 0 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*r* 98.0%

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

   3. associate-/r* 98.1%

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

10. Simplified 98.1%

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

11. Final simplification 98.1%

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

Alternative 9: 98.0% 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* 99.4%

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

   3. associate-/l/ 99.4%

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

   4. sub-neg 99.4%

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

   5. +-commutative 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-neg-in 99.4%

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

   8. fma-def 99.4%

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

   9. metadata-eval 99.4%

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

   10. sub-neg 99.4%

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

   11. distribute-rgt-in 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in v around 0 97.5%

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

5. Final simplification 97.5%

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

Reproduce

herbie shell --seed 2023278 
(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)))))