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

Percentage Accurate: 99.3% → 99.5%

Time: 3.0s

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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, t_] := N[(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] * N[Sqrt[N[Power[N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision], -1.0], $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

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

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-*.f64 99.3

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in t around 0

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

9. Applied rewrites 99.5%

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

Alternative 2: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-*.f64 99.3

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

6. Applied rewrites 99.3%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

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

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-fma.f64 99.3

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

8. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 99.3

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

10. Applied rewrites 99.3%

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

Alternative 3: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, t_] := N[(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] * 1.0), $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

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-*.f64 99.3

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in t around 0

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in v around 0

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

12. Applied rewrites 98.5%

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

Alternative 4: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-*.f64 99.3

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in v around 0

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

   1. associate-*r* N/A

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

   2. sqrt-prod N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 98.4

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

9. Applied rewrites 98.4%

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

    1. lift--.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. pow2 N/A

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

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

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

    6. metadata-eval N/A

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

    7. +-commutative N/A

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

    8. *-commutative N/A

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

    9. lower-fma.f64 N/A

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

    10. pow2 N/A

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

    11. lift-*.f64 98.4

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

11. Applied rewrites 98.4%

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

Alternative 5: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, t_] := N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 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. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 98.4

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

6. Applied rewrites 98.4%

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

Alternative 6: 98.3% 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

7. Applied rewrites 98.3%

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

Alternative 7: 98.3% accurate, 2.4× speedup

Math

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

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

7. Applied rewrites 98.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. *-commutative N/A

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

   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-sqrt.f64 98.3

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

9. Applied rewrites 98.3%

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

    1. lift-*.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-sqrt.f64 N/A

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

    5. associate-*l* N/A

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

    6. lower-*.f64 N/A

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

    7. lift-PI.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. lift-sqrt.f64 98.3

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

11. Applied rewrites 98.3%

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

Reproduce

herbie shell --seed 2025102 
(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)))))