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

Percentage Accurate: 99.2% → 99.8%

Time: 4.1s

Alternatives: 7

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: 99.2% 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: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

3. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-fma.f64 N/A

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

   4. add-flip N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. sub-negate-rev N/A

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

   9. lift-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. add-flip N/A

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

   12. metadata-eval N/A

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

   13. sub-negate-rev N/A

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

   14. lift--.f64 N/A

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

   15. div-sub N/A

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

5. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. distribute-lft1-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. metadata-eval 99.8

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

7. Applied rewrites 99.8%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sub-div N/A

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

   5. sub-negate-rev N/A

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

   6. lift-fma.f64 N/A

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

   7. add-flip N/A

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

   8. metadata-eval N/A

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

   9. sub-negate-rev N/A

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

   10. sub-negate-rev N/A

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

   11. metadata-eval N/A

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

   12. add-flip N/A

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

   13. lift-fma.f64 N/A

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

   14. frac-2neg-rev N/A

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

   15. lower-/.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. associate-*r* N/A

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

   20. lower-*.f64 N/A

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

   21. lower-*.f64 N/A

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

   22. lift-fma.f64 N/A

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

   23. add-flip N/A

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

9. Applied rewrites 99.8%

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

Alternative 2: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. lift--.f64 N/A

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

   7. sub-flip N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. remove-double-neg N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

3. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 99.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 99.2

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. distribute-lft1-in N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. metadata-eval N/A

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

   17. lift-*.f64 99.2

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

7. Applied rewrites 99.2%

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

Alternative 3: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. lift--.f64 N/A

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

   7. sub-flip N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. remove-double-neg N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

3. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 99.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. sqrt-prod N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. metadata-eval N/A

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

   11. fp-cancel-sub-sign-inv N/A

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

   12. sqrt-prod N/A

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

   13. *-commutative N/A

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

7. Applied rewrites 99.2%

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

Alternative 4: 98.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in v around 0

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

6. Applied rewrites 98.5%

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

Alternative 5: 98.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 N/A

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

   4. lower-sqrt.f64 98.5

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

6. Applied rewrites 98.5%

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

Alternative 6: 98.1% accurate, 3.3× speedup

Math

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

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

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-sqrt.f64 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 98.1

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 98.1

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

6. Applied rewrites 98.1%

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

Alternative 7: 98.0% accurate, 3.4× 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 99.2%

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

2. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-sqrt.f64 98.0

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

4. Applied rewrites 98.0%

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

Reproduce

herbie shell --seed 2025155 
(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)))))