(- PI (* (* (* z0 z0) PI) (* (* PI PI) 8333333333333333/50000000000000000)))

Percentage Accurate: 99.6% → 99.9%

Time: 1.2s

Alternatives: 2

Speedup: 1.5×

• 75.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\pi - \left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.16666666666666666\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (- PI (* (* (* z0 z0) PI) (* (* PI PI) 0.16666666666666666))))

C

double code(double z0) {
	return ((double) M_PI) - (((z0 * z0) * ((double) M_PI)) * ((((double) M_PI) * ((double) M_PI)) * 0.16666666666666666));
}

Java

public static double code(double z0) {
	return Math.PI - (((z0 * z0) * Math.PI) * ((Math.PI * Math.PI) * 0.16666666666666666));
}

Python

def code(z0):
	return math.pi - (((z0 * z0) * math.pi) * ((math.pi * math.pi) * 0.16666666666666666))

Julia

function code(z0)
	return Float64(pi - Float64(Float64(Float64(z0 * z0) * pi) * Float64(Float64(pi * pi) * 0.16666666666666666)))
end

MATLAB

function tmp = code(z0)
	tmp = pi - (((z0 * z0) * pi) * ((pi * pi) * 0.16666666666666666));
end

Wolfram

code[z0_] := N[(Pi - N[(N[(N[(z0 * z0), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\pi - \left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.16666666666666666\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (- PI (* (* (* z0 z0) PI) (* (* PI PI) 0.16666666666666666))))

C

double code(double z0) {
	return ((double) M_PI) - (((z0 * z0) * ((double) M_PI)) * ((((double) M_PI) * ((double) M_PI)) * 0.16666666666666666));
}

Java

public static double code(double z0) {
	return Math.PI - (((z0 * z0) * Math.PI) * ((Math.PI * Math.PI) * 0.16666666666666666));
}

Python

def code(z0):
	return math.pi - (((z0 * z0) * math.pi) * ((math.pi * math.pi) * 0.16666666666666666))

Julia

function code(z0)
	return Float64(pi - Float64(Float64(Float64(z0 * z0) * pi) * Float64(Float64(pi * pi) * 0.16666666666666666)))
end

MATLAB

function tmp = code(z0)
	tmp = pi - (((z0 * z0) * pi) * ((pi * pi) * 0.16666666666666666));
end

Wolfram

code[z0_] := N[(Pi - N[(N[(N[(z0 * z0), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.5× speedup

Math

\[\left(\left(z0 \cdot z0\right) \cdot \pi - 1.909859317102744\right) \cdot -1.6449340668482264 \]

FPCore

(FPCore (z0)
  :precision binary64
  (* (- (* (* z0 z0) PI) 1.909859317102744) -1.6449340668482264))

C

double code(double z0) {
	return (((z0 * z0) * ((double) M_PI)) - 1.909859317102744) * -1.6449340668482264;
}

Java

public static double code(double z0) {
	return (((z0 * z0) * Math.PI) - 1.909859317102744) * -1.6449340668482264;
}

Python

def code(z0):
	return (((z0 * z0) * math.pi) - 1.909859317102744) * -1.6449340668482264

Julia

function code(z0)
	return Float64(Float64(Float64(Float64(z0 * z0) * pi) - 1.909859317102744) * -1.6449340668482264)
end

MATLAB

function tmp = code(z0)
	tmp = (((z0 * z0) * pi) - 1.909859317102744) * -1.6449340668482264;
end

Wolfram

code[z0_] := N[(N[(N[(N[(z0 * z0), $MachinePrecision] * Pi), $MachinePrecision] - 1.909859317102744), $MachinePrecision] * -1.6449340668482264), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\pi - \left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.16666666666666666\right) \]

2. Evaluated real constant 99.9%

   \[\leadsto \pi - \left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \color{blue}{1.6449340668482264} \]
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\pi - \left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \frac{7408124450506707}{4503599627370496}} \]

   2. lift-*.f64 N/A

      \[\leadsto \pi - \color{blue}{\left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \frac{7408124450506707}{4503599627370496}} \]

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

      \[\leadsto \color{blue}{\pi + \left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{7408124450506707}{4503599627370496}} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{7408124450506707}{4503599627370496} + \pi} \]

   5. sum-to-mult N/A

      \[\leadsto \color{blue}{\left(1 + \frac{\pi}{\left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{7408124450506707}{4503599627370496}}\right) \cdot \left(\left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{7408124450506707}{4503599627370496}\right)} \]

   6. lower-unsound-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 + \frac{\pi}{\left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{7408124450506707}{4503599627370496}}\right) \cdot \left(\left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{7408124450506707}{4503599627370496}\right)} \]

   7. lower-unsound-+.f64 N/A

      \[\leadsto \color{blue}{\left(1 + \frac{\pi}{\left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{7408124450506707}{4503599627370496}}\right)} \cdot \left(\left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{7408124450506707}{4503599627370496}\right) \]

   8. lower-unsound-/.f64 N/A

      \[\leadsto \left(1 + \color{blue}{\frac{\pi}{\left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{7408124450506707}{4503599627370496}}}\right) \cdot \left(\left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{7408124450506707}{4503599627370496}\right) \]

   9. *-commutative N/A

      \[\leadsto \left(1 + \frac{\pi}{\color{blue}{\frac{7408124450506707}{4503599627370496} \cdot \left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right)}}\right) \cdot \left(\left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{7408124450506707}{4503599627370496}\right) \]

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

       \[\leadsto \left(1 + \frac{\pi}{\color{blue}{\mathsf{neg}\left(\frac{7408124450506707}{4503599627370496} \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)\right)}}\right) \cdot \left(\left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{7408124450506707}{4503599627370496}\right) \]

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

       \[\leadsto \left(1 + \frac{\pi}{\color{blue}{\left(\mathsf{neg}\left(\frac{7408124450506707}{4503599627370496}\right)\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)}}\right) \cdot \left(\left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{7408124450506707}{4503599627370496}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \left(1 + \frac{\pi}{\color{blue}{\left(\mathsf{neg}\left(\frac{7408124450506707}{4503599627370496}\right)\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)}}\right) \cdot \left(\left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{7408124450506707}{4503599627370496}\right) \]

   13. metadata-eval N/A

       \[\leadsto \left(1 + \frac{\pi}{\color{blue}{\frac{-7408124450506707}{4503599627370496}} \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{7408124450506707}{4503599627370496}\right) \]

   14. *-commutative N/A

       \[\leadsto \left(1 + \frac{\pi}{\frac{-7408124450506707}{4503599627370496} \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) \cdot \color{blue}{\left(\frac{7408124450506707}{4503599627370496} \cdot \left(\mathsf{neg}\left(\left(z0 \cdot z0\right) \cdot \pi\right)\right)\right)} \]

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

       \[\leadsto \left(1 + \frac{\pi}{\frac{-7408124450506707}{4503599627370496} \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{7408124450506707}{4503599627370496} \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)\right)\right)} \]

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

       \[\leadsto \left(1 + \frac{\pi}{\frac{-7408124450506707}{4503599627370496} \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{7408124450506707}{4503599627370496}\right)\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)\right)} \]

   17. lower-*.f64 N/A

       \[\leadsto \left(1 + \frac{\pi}{\frac{-7408124450506707}{4503599627370496} \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{7408124450506707}{4503599627370496}\right)\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)\right)} \]

4. Applied rewrites 74.8%

   \[\leadsto \color{blue}{\left(1 + \frac{\pi}{-1.6449340668482264 \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) \cdot \left(-1.6449340668482264 \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 + \frac{\pi}{\frac{-7408124450506707}{4503599627370496} \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) \cdot \left(\frac{-7408124450506707}{4503599627370496} \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 + \frac{\pi}{\frac{-7408124450506707}{4503599627370496} \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) \cdot \color{blue}{\left(\frac{-7408124450506707}{4503599627370496} \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \left(1 + \frac{\pi}{\frac{-7408124450506707}{4503599627370496} \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) \cdot \color{blue}{\left(\left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \frac{-7408124450506707}{4503599627370496}\right)} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(1 + \frac{\pi}{\frac{-7408124450506707}{4503599627370496} \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)}\right) \cdot \left(\left(z0 \cdot z0\right) \cdot \pi\right)\right) \cdot \frac{-7408124450506707}{4503599627370496}} \]

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\left(z0 \cdot z0\right) \cdot \pi - \pi \cdot 0.6079271018540267\right) \cdot -1.6449340668482264} \]

7. Evaluated real constant 99.9%

   \[\leadsto \left(\left(z0 \cdot z0\right) \cdot \pi - \color{blue}{1.909859317102744}\right) \cdot -1.6449340668482264 \]
8. Add Preprocessing

Alternative 2: 50.6% accurate, 29.0× speedup

Math

\[\pi \]

FPCore

(FPCore (z0)
  :precision binary64
  PI)

C

double code(double z0) {
	return (double) M_PI;
}

Java

public static double code(double z0) {
	return Math.PI;
}

Python

def code(z0):
	return math.pi

Julia

function code(z0)
	return pi
end

MATLAB

function tmp = code(z0)
	tmp = pi;
end

Wolfram

code[z0_] := Pi

Derivation

1. Initial program 99.6%

   \[\pi - \left(\left(z0 \cdot z0\right) \cdot \pi\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.16666666666666666\right) \]

2. Taylor expanded in z0 around 0

   \[\leadsto \color{blue}{\pi} \]
3. Step-by-step derivation

   1. lower-PI.f64 50.6%

      \[\leadsto \pi \]

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\pi} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z0)
  :name "(- PI (* (* (* z0 z0) PI) (* (* PI PI) 8333333333333333/50000000000000000)))"
  :precision binary64
  (- PI (* (* (* z0 z0) PI) (* (* PI PI) 0.16666666666666666))))