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

Percentage Accurate: 99.3% → 99.4%

Time: 3.1s

Alternatives: 5

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, t_] := N[(N[(N[(5.0 * N[(v * v), $MachinePrecision] + -1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sqrt[N[(-3.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $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. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.3

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

   7. lift--.f64 N/A

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

   8. sub-flip N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-lft-neg-out N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval 99.3

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

3. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

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

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

   6. metadata-eval N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift--.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. sub-flip N/A

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

   13. metadata-eval N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 N/A

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

   16. lift--.f64 N/A

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

   17. sub-negate-rev N/A

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

   18. sub-flip N/A

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

   19. lift-*.f64 N/A

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

   20. metadata-eval N/A

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

   21. lower-fma.f64 99.4

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

5. Applied rewrites 99.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqrt-prod N/A

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

   4. lift-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. metadata-eval N/A

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

   15. +-commutative N/A

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

   16. lift-fma.f64 99.4

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

7. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, t_] := N[(N[(N[(5.0 * N[(v * v), $MachinePrecision] + -1.0), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[N[(N[(-3.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision] * 2.0), $MachinePrecision]], $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. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.3

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

   7. lift--.f64 N/A

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

   8. sub-flip N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-lft-neg-out N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval 99.3

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

3. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

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

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

   6. metadata-eval N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift--.f64 N/A

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

   11. sub-negate-rev N/A

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

   12. sub-flip N/A

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

   13. metadata-eval N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 N/A

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

   16. lift--.f64 N/A

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

   17. sub-negate-rev N/A

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

   18. sub-flip N/A

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

   19. lift-*.f64 N/A

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

   20. metadata-eval N/A

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

   21. lower-fma.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-negate-rev N/A

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

   6. sub-flip N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

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

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

3. Applied rewrites 99.4%

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

Alternative 4: 98.8% accurate, 3.3× speedup

Math

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

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

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-sqrt.f64 98.4

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

4. Applied rewrites 98.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 98.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.8

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

6. Applied rewrites 98.8%

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

Alternative 5: 98.4% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-sqrt.f64 98.4

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

4. Applied rewrites 98.4%

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

Reproduce

herbie shell --seed 2025151 
(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)))))