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

Percentage Accurate: 99.3% → 99.8%

Time: 3.0s

Alternatives: 8

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: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

5. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

7. Applied rewrites 99.8%

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

Alternative 2: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

5. Applied rewrites 99.4%

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

Alternative 3: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 99.3

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

4. Applied rewrites 99.3%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

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

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

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 99.3

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

6. Applied rewrites 99.3%

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

Alternative 4: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. sqrt-prod N/A

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

   7. pow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. pow2 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. pow2 N/A

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

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

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

   14. metadata-eval N/A

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

   15. +-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. pow2 N/A

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

   18. lift-*.f64 99.3

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in v around 0

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

9. Applied rewrites 98.5%

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

Alternative 5: 98.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. sqrt-prod N/A

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

   7. pow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. pow2 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. pow2 N/A

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

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

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

   14. metadata-eval N/A

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

   15. +-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. pow2 N/A

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

   18. lift-*.f64 99.3

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in v around 0

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

9. Applied rewrites 98.5%

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

10. Taylor expanded in v around 0

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

    1. *-commutative N/A

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

    2. lift-sqrt.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-PI.f64 98.5

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

12. Applied rewrites 98.5%

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

Alternative 6: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 98.4

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

4. Applied rewrites 98.4%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

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

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

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 98.4

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

6. Applied rewrites 98.4%

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

Alternative 7: 98.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in v around 0

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

   1. pow2 98.4

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

   2. fp-cancel-sub-sign-inv 98.4

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

   3. metadata-eval 98.4

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

   4. +-commutative 98.4

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

   5. pow2 98.4

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 98.4

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

7. Applied rewrites 98.4%

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

Alternative 8: 98.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in v around 0

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

   1. pow2 98.4

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

   2. fp-cancel-sub-sign-inv 98.4

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

   3. metadata-eval 98.4

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

   4. +-commutative 98.4

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

   5. pow2 98.4

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 98.4

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

7. Applied rewrites 98.4%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-PI.f64 N/A

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

   17. lift-sqrt.f64 98.3

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

9. Applied rewrites 98.3%

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

Reproduce

herbie shell --seed 2025096 
(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)))))