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

Percentage Accurate: 99.3% → 99.6%

Time: 8.8s

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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. associate-/r* 99.3%

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

   3. associate-/l/ 99.3%

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

   4. sub-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. *-commutative 99.3%

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

   7. distribute-rgt-neg-in 99.3%

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

   8. fma-def 99.3%

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

   9. metadata-eval 99.3%

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

   10. sub-neg 99.3%

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

   11. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in v around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. fma-def 99.3%

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

   4. unpow2 99.3%

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

   5. times-frac 99.3%

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

   6. associate-/r* 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. associate-/r* 99.3%

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

   3. associate-/l/ 99.3%

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

   4. sub-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. *-commutative 99.3%

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

   7. distribute-rgt-neg-in 99.3%

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

   8. fma-def 99.3%

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

   9. metadata-eval 99.3%

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

   10. sub-neg 99.3%

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

   11. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in t around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. unpow2 99.3%

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

   4. fma-udef 99.3%

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

   5. fma-udef 99.3%

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

   6. associate-*r* 99.3%

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

   7. fma-udef 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in t around 0 99.3%

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

   1. *-commutative 99.3%

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

   2. unpow2 99.3%

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

9. Simplified 99.3%

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

    1. expm1-log1p-u 72.0%

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

    2. expm1-udef 23.1%

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

11. Applied egg-rr 23.1%

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

    1. expm1-def 72.0%

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

    2. expm1-log1p 99.3%

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

    3. associate-/l/ 99.5%

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

    4. *-commutative 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. associate-/r* 99.3%

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

   3. associate-/l/ 99.3%

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

   4. sub-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. *-commutative 99.3%

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

   7. distribute-rgt-neg-in 99.3%

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

   8. fma-def 99.3%

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

   9. metadata-eval 99.3%

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

   10. sub-neg 99.3%

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

   11. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in t around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. unpow2 99.3%

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

   4. fma-udef 99.3%

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

   5. fma-udef 99.3%

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

   6. associate-*r* 99.3%

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

   7. fma-udef 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in t around 0 99.3%

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

   1. *-commutative 99.3%

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

   2. unpow2 99.3%

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

9. Simplified 99.3%

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

10. Taylor expanded in t around 0 99.3%

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

    1. *-commutative 99.3%

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

    2. unpow2 99.3%

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

    3. associate-*r* 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. associate-/r* 99.3%

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

   3. associate-/l/ 99.3%

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

   4. sub-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. *-commutative 99.3%

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

   7. distribute-rgt-neg-in 99.3%

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

   8. fma-def 99.3%

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

   9. metadata-eval 99.3%

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

   10. sub-neg 99.3%

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

   11. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in t around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. unpow2 99.3%

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

   4. fma-udef 99.3%

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

   5. fma-udef 99.3%

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

   6. associate-*r* 99.3%

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

   7. fma-udef 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in t around 0 99.3%

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

   1. *-commutative 99.3%

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

   2. unpow2 99.3%

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

9. Simplified 99.3%

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

10. Taylor expanded in v around 0 99.3%

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

    1. *-commutative 99.3%

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

    2. unpow2 99.3%

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

    3. associate-*r* 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

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

Alternative 6: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. associate-/r* 99.3%

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

   3. associate-/l/ 99.3%

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

   4. sub-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. *-commutative 99.3%

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

   7. distribute-rgt-neg-in 99.3%

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

   8. fma-def 99.3%

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

   9. metadata-eval 99.3%

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

   10. sub-neg 99.3%

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

   11. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in v around 0 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-/r* 97.6%

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

6. Simplified 97.6%

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

7. Taylor expanded in v around 0 97.7%

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

   1. expm1-log1p-u 70.7%

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

   2. expm1-udef 22.7%

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

   3. associate-/l/ 22.7%

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

   4. associate-/l/ 22.7%

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

   5. associate-/r* 22.7%

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

9. Applied egg-rr 22.7%

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

    1. expm1-def 70.6%

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

    2. expm1-log1p 97.5%

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

    3. associate-/l/ 97.6%

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

11. Simplified 97.6%

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

12. Final simplification 97.6%

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

Alternative 7: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. associate-/r* 99.3%

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

   3. associate-/l/ 99.3%

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

   4. sub-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. *-commutative 99.3%

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

   7. distribute-rgt-neg-in 99.3%

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

   8. fma-def 99.3%

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

   9. metadata-eval 99.3%

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

   10. sub-neg 99.3%

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

   11. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in t around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. unpow2 99.3%

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

   4. fma-udef 99.3%

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

   5. fma-udef 99.3%

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

   6. associate-*r* 99.3%

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

   7. fma-udef 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in t around 0 99.3%

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

   1. *-commutative 99.3%

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

   2. unpow2 99.3%

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

9. Simplified 99.3%

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

    1. expm1-log1p-u 72.0%

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

    2. expm1-udef 23.1%

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

11. Applied egg-rr 23.1%

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

    1. expm1-def 72.0%

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

    2. expm1-log1p 99.3%

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

    3. associate-/l/ 99.5%

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

    4. *-commutative 99.5%

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

13. Simplified 99.5%

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

14. Taylor expanded in v around 0 97.7%

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

15. Final simplification 97.7%

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

Alternative 8: 98.4% 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 99.3%

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

   1. associate-*l* 99.3%

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

   2. associate-/r* 99.3%

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

   3. associate-/l/ 99.3%

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

   4. sub-neg 99.3%

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

   5. +-commutative 99.3%

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

   6. *-commutative 99.3%

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

   7. distribute-rgt-neg-in 99.3%

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

   8. fma-def 99.3%

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

   9. metadata-eval 99.3%

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

   10. sub-neg 99.3%

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

   11. distribute-rgt-in 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in v around 0 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-/r* 97.6%

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

6. Simplified 97.6%

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

7. Taylor expanded in v around 0 97.7%

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

8. Taylor expanded in v around 0 97.4%

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

   1. expm1-log1p-u 68.8%

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

   2. expm1-udef 26.3%

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

10. Applied egg-rr 26.3%

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

    1. expm1-def 68.8%

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

    2. expm1-log1p 97.4%

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

    3. *-commutative 97.4%

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

    4. associate-*l* 97.6%

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

12. Simplified 97.6%

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

13. Final simplification 97.6%

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

Alternative 9: 97.9% accurate, 1.1× speedup

Math

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

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(sqrt(0.5) / Float64(t * pi))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. associate-*l* 99.3%

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

   2. sub-neg 99.3%

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

   3. distribute-lft-in 99.3%

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

   4. metadata-eval 99.3%

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

   5. *-commutative 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

   7. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in v around 0 97.2%

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

5. Final simplification 97.2%

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

Reproduce

herbie shell --seed 2023213 
(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)))))