xkcd217 (a numerical coincidence)

Percentage Accurate: 77.8% → 83.9%

Time: 2.0min

Alternatives: 1

Speedup: 0.4×

Specification

Math

\[20 - \left(e^{\pi} - \pi\right) \]

FPCore

(FPCore ()
  :precision binary64
  (- 20 (- (exp PI) PI)))

C

double code() {
	return 20.0 - (exp(((double) M_PI)) - ((double) M_PI));
}

Java

public static double code() {
	return 20.0 - (Math.exp(Math.PI) - Math.PI);
}

Python

def code():
	return 20.0 - (math.exp(math.pi) - math.pi)

Julia

function code()
	return Float64(20.0 - Float64(exp(pi) - pi))
end

MATLAB

function tmp = code()
	tmp = 20.0 - (exp(pi) - pi);
end

Wolfram

code[] := N[(20 - N[(N[Exp[Pi], $MachinePrecision] - Pi), $MachinePrecision]), $MachinePrecision]

Initial Program: 77.8% accurate, 1.0× speedup

Math

\[20 - \left(e^{\pi} - \pi\right) \]

FPCore

(FPCore ()
  :precision binary64
  (- 20 (- (exp PI) PI)))

C

double code() {
	return 20.0 - (exp(((double) M_PI)) - ((double) M_PI));
}

Java

public static double code() {
	return 20.0 - (Math.exp(Math.PI) - Math.PI);
}

Python

def code():
	return 20.0 - (math.exp(math.pi) - math.pi)

Julia

function code()
	return Float64(20.0 - Float64(exp(pi) - pi))
end

MATLAB

function tmp = code()
	tmp = 20.0 - (exp(pi) - pi);
end

Wolfram

code[] := N[(20 - N[(N[Exp[Pi], $MachinePrecision] - Pi), $MachinePrecision]), $MachinePrecision]

Alternative 1: 83.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{-1}{20} \cdot e^{\pi}\\ \left(\left(1 + \frac{1}{t\_0}\right) \cdot t\_0\right) \cdot 20 + \pi \end{array} \]

FPCore

(FPCore ()
  :precision binary64
  (let* ((t_0 (* -1/20 (exp PI))))
  (+ (* (* (+ 1 (/ 1 t_0)) t_0) 20) PI)))

C

double code() {
	double t_0 = -0.05 * exp(((double) M_PI));
	return (((1.0 + (1.0 / t_0)) * t_0) * 20.0) + ((double) M_PI);
}

Java

public static double code() {
	double t_0 = -0.05 * Math.exp(Math.PI);
	return (((1.0 + (1.0 / t_0)) * t_0) * 20.0) + Math.PI;
}

Python

def code():
	t_0 = -0.05 * math.exp(math.pi)
	return (((1.0 + (1.0 / t_0)) * t_0) * 20.0) + math.pi

Julia

function code()
	t_0 = Float64(-0.05 * exp(pi))
	return Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / t_0)) * t_0) * 20.0) + pi)
end

MATLAB

function tmp = code()
	t_0 = -0.05 * exp(pi);
	tmp = (((1.0 + (1.0 / t_0)) * t_0) * 20.0) + pi;
end

Wolfram

code[] := Block[{t$95$0 = N[(-1/20 * N[Exp[Pi], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(1 + N[(1 / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * 20), $MachinePrecision] + Pi), $MachinePrecision]]

Derivation

1. Initial program 77.8%

   \[20 - \left(e^{\pi} - \pi\right) \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{20 - \left(e^{\pi} - \pi\right)} \]

   2. sub-to-mult N/A

      \[\leadsto \color{blue}{\left(1 - \frac{e^{\pi} - \pi}{20}\right) \cdot 20} \]

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

      \[\leadsto \color{blue}{\left(1 - \frac{e^{\pi} - \pi}{20}\right) \cdot 20} \]

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

      \[\leadsto \color{blue}{\left(1 - \frac{e^{\pi} - \pi}{20}\right)} \cdot 20 \]

   5. lower-unsound-/.f64 78.3%

      \[\leadsto \left(1 - \color{blue}{\frac{e^{\pi} - \pi}{20}}\right) \cdot 20 \]

3. Applied rewrites 78.3%

   \[\leadsto \color{blue}{\left(1 - \frac{e^{\pi} - \pi}{20}\right) \cdot 20} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\frac{e^{\pi} - \pi}{20}}\right) \cdot 20 \]

   2. lift--.f64 N/A

      \[\leadsto \left(1 - \frac{\color{blue}{e^{\pi} - \pi}}{20}\right) \cdot 20 \]

   3. div-sub N/A

      \[\leadsto \left(1 - \color{blue}{\left(\frac{e^{\pi}}{20} - \frac{\pi}{20}\right)}\right) \cdot 20 \]

   4. lower--.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\left(\frac{e^{\pi}}{20} - \frac{\pi}{20}\right)}\right) \cdot 20 \]

   5. mult-flip N/A

      \[\leadsto \left(1 - \left(\color{blue}{e^{\pi} \cdot \frac{1}{20}} - \frac{\pi}{20}\right)\right) \cdot 20 \]

   6. lower-*.f64 N/A

      \[\leadsto \left(1 - \left(\color{blue}{e^{\pi} \cdot \frac{1}{20}} - \frac{\pi}{20}\right)\right) \cdot 20 \]

   7. metadata-eval N/A

      \[\leadsto \left(1 - \left(e^{\pi} \cdot \color{blue}{\frac{1}{20}} - \frac{\pi}{20}\right)\right) \cdot 20 \]

   8. mult-flip N/A

      \[\leadsto \left(1 - \left(e^{\pi} \cdot \frac{1}{20} - \color{blue}{\pi \cdot \frac{1}{20}}\right)\right) \cdot 20 \]

   9. lower-*.f64 N/A

      \[\leadsto \left(1 - \left(e^{\pi} \cdot \frac{1}{20} - \color{blue}{\pi \cdot \frac{1}{20}}\right)\right) \cdot 20 \]

   10. metadata-eval 80.6%

       \[\leadsto \left(1 - \left(e^{\pi} \cdot \frac{1}{20} - \pi \cdot \color{blue}{\frac{1}{20}}\right)\right) \cdot 20 \]

5. Applied rewrites 80.6%

   \[\leadsto \left(1 - \color{blue}{\left(e^{\pi} \cdot \frac{1}{20} - \pi \cdot \frac{1}{20}\right)}\right) \cdot 20 \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \left(e^{\pi} \cdot \frac{1}{20} - \pi \cdot \frac{1}{20}\right)\right) \cdot 20} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{20 \cdot \left(1 - \left(e^{\pi} \cdot \frac{1}{20} - \pi \cdot \frac{1}{20}\right)\right)} \]

   3. lift--.f64 N/A

      \[\leadsto 20 \cdot \color{blue}{\left(1 - \left(e^{\pi} \cdot \frac{1}{20} - \pi \cdot \frac{1}{20}\right)\right)} \]

   4. lift--.f64 N/A

      \[\leadsto 20 \cdot \left(1 - \color{blue}{\left(e^{\pi} \cdot \frac{1}{20} - \pi \cdot \frac{1}{20}\right)}\right) \]

   5. associate--r- N/A

      \[\leadsto 20 \cdot \color{blue}{\left(\left(1 - e^{\pi} \cdot \frac{1}{20}\right) + \pi \cdot \frac{1}{20}\right)} \]

   6. distribute-lft-in N/A

      \[\leadsto \color{blue}{20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + 20 \cdot \left(\pi \cdot \frac{1}{20}\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto 20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + 20 \cdot \color{blue}{\left(\pi \cdot \frac{1}{20}\right)} \]

   8. *-commutative N/A

      \[\leadsto 20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + 20 \cdot \color{blue}{\left(\frac{1}{20} \cdot \pi\right)} \]

   9. associate-*r* N/A

      \[\leadsto 20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + \color{blue}{\left(20 \cdot \frac{1}{20}\right) \cdot \pi} \]

   10. metadata-eval N/A

       \[\leadsto 20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + \color{blue}{1} \cdot \pi \]

   11. lift-PI.f64 N/A

       \[\leadsto 20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + 1 \cdot \color{blue}{\mathsf{PI}\left(\right)} \]

   12. add-log-exp N/A

       \[\leadsto 20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + 1 \cdot \color{blue}{\log \left(e^{\mathsf{PI}\left(\right)}\right)} \]

   13. lift-PI.f64 N/A

       \[\leadsto 20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + 1 \cdot \log \left(e^{\color{blue}{\pi}}\right) \]

   14. lift-exp.f64 N/A

       \[\leadsto 20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + 1 \cdot \log \color{blue}{\left(e^{\pi}\right)} \]

   15. log-pow-rev N/A

       \[\leadsto 20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + \color{blue}{\log \left({\left(e^{\pi}\right)}^{1}\right)} \]

   16. metadata-eval N/A

       \[\leadsto 20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + \log \left({\left(e^{\pi}\right)}^{\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}}\right) \]

   17. pow-flip N/A

       \[\leadsto 20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + \log \color{blue}{\left(\frac{1}{{\left(e^{\pi}\right)}^{-1}}\right)} \]

   18. inv-pow N/A

       \[\leadsto 20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + \log \left(\frac{1}{\color{blue}{\frac{1}{e^{\pi}}}}\right) \]

   19. lift-exp.f64 N/A

       \[\leadsto 20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + \log \left(\frac{1}{\frac{1}{\color{blue}{e^{\pi}}}}\right) \]

   20. lift-PI.f64 N/A

       \[\leadsto 20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + \log \left(\frac{1}{\frac{1}{e^{\color{blue}{\mathsf{PI}\left(\right)}}}}\right) \]

   21. add-cube-cbrt N/A

       \[\leadsto 20 \cdot \left(1 - e^{\pi} \cdot \frac{1}{20}\right) + \log \left(\frac{1}{\frac{1}{e^{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}}}\right) \]

7. Applied rewrites 80.6%

   \[\leadsto \color{blue}{\left(1 - \frac{1}{20} \cdot e^{\pi}\right) \cdot 20 + \pi} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \frac{1}{20} \cdot e^{\pi}\right)} \cdot 20 + \pi \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\frac{1}{20} \cdot e^{\pi}}\right) \cdot 20 + \pi \]

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

      \[\leadsto \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{20}\right)\right) \cdot e^{\pi}\right)} \cdot 20 + \pi \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{20}\right)\right) \cdot e^{\pi} + 1\right)} \cdot 20 + \pi \]

   5. sum-to-mult N/A

      \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{\left(\mathsf{neg}\left(\frac{1}{20}\right)\right) \cdot e^{\pi}}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{20}\right)\right) \cdot e^{\pi}\right)\right)} \cdot 20 + \pi \]

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

      \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{\left(\mathsf{neg}\left(\frac{1}{20}\right)\right) \cdot e^{\pi}}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{20}\right)\right) \cdot e^{\pi}\right)\right)} \cdot 20 + \pi \]

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

      \[\leadsto \left(\color{blue}{\left(1 + \frac{1}{\left(\mathsf{neg}\left(\frac{1}{20}\right)\right) \cdot e^{\pi}}\right)} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{20}\right)\right) \cdot e^{\pi}\right)\right) \cdot 20 + \pi \]

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

      \[\leadsto \left(\left(1 + \color{blue}{\frac{1}{\left(\mathsf{neg}\left(\frac{1}{20}\right)\right) \cdot e^{\pi}}}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{20}\right)\right) \cdot e^{\pi}\right)\right) \cdot 20 + \pi \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\left(1 + \frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{20}\right)\right) \cdot e^{\pi}}}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{20}\right)\right) \cdot e^{\pi}\right)\right) \cdot 20 + \pi \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(1 + \frac{1}{\color{blue}{\frac{-1}{20}} \cdot e^{\pi}}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{20}\right)\right) \cdot e^{\pi}\right)\right) \cdot 20 + \pi \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\left(1 + \frac{1}{\frac{-1}{20} \cdot e^{\pi}}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{20}\right)\right) \cdot e^{\pi}\right)}\right) \cdot 20 + \pi \]

   12. metadata-eval 83.9%

       \[\leadsto \left(\left(1 + \frac{1}{\frac{-1}{20} \cdot e^{\pi}}\right) \cdot \left(\color{blue}{\frac{-1}{20}} \cdot e^{\pi}\right)\right) \cdot 20 + \pi \]

9. Applied rewrites 83.9%

   \[\leadsto \color{blue}{\left(\left(1 + \frac{1}{\frac{-1}{20} \cdot e^{\pi}}\right) \cdot \left(\frac{-1}{20} \cdot e^{\pi}\right)\right)} \cdot 20 + \pi \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore ()
  :name "xkcd217 (a numerical coincidence)"
  :precision binary64
  (- 20 (- (exp PI) PI)))