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

Percentage Accurate: 99.3% → 99.5%

Time: 13.7s

Alternatives: 10

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. *-rgt-identity N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.6%

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

Alternative 2: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. distribute-lft-neg-in N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. *-lowering-*.f64 N/A

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

   16. PI-lowering-PI.f64 99.2

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

4. Applied egg-rr 99.2%

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. +-commutative N/A

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

   7. associate-*r* N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-lowering-*.f64 99.2

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

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

Alternative 3: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. /-lowering-/.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. *-lowering-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. associate-*l* N/A

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

   13. associate-*l* N/A

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

   14. *-lowering-*.f64 N/A

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

   15. PI-lowering-PI.f64 N/A

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

   16. associate-*r* N/A

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

4. Applied egg-rr 99.1%

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

5. Final simplification 99.1%

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

Alternative 4: 98.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. *-rgt-identity N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around 0

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

   1. PI-lowering-PI.f64 98.4

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

7. Simplified 98.4%

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

Alternative 5: 98.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. *-rgt-identity N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around 0

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

   1. PI-lowering-PI.f64 98.4

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

7. Simplified 98.4%

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

   1. associate-/r* N/A

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

   2. frac-times N/A

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

   3. div-inv N/A

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

   4. /-lowering-/.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. PI-lowering-PI.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. sqrt-lowering-sqrt.f64 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 98.4

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

9. Applied egg-rr 98.4%

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

Alternative 6: 98.6% accurate, 1.6× speedup

Math

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

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(Float64(1.0 / 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[(N[(1.0 / Pi), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. PI-lowering-PI.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 97.9

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

5. Simplified 97.9%

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

   1. associate-/r* N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. div-inv N/A

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

   5. *-lowering-*.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. PI-lowering-PI.f64 N/A

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

   9. sqrt-lowering-sqrt.f64 N/A

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

   10. /-lowering-/.f64 98.4

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

7. Applied egg-rr 98.4%

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

Alternative 7: 98.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. PI-lowering-PI.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 97.9

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

5. Simplified 97.9%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. PI-lowering-PI.f64 98.1

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

7. Applied egg-rr 98.1%

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

Alternative 8: 98.5% accurate, 2.4× 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 99.1%

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

3. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. PI-lowering-PI.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 97.9

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

5. Simplified 97.9%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. PI-lowering-PI.f64 97.9

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

7. Applied egg-rr 97.9%

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

8. Final simplification 97.9%

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

Alternative 9: 98.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. PI-lowering-PI.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 97.9

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

5. Simplified 97.9%

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

Alternative 10: 97.9% accurate, 2.8× 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 99.1%

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

   1. *-rgt-identity N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 97.4

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

7. Simplified 97.4%

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

Reproduce

herbie shell --seed 2024199 
(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)))))