Average Error: 16.5 → 12.6

Time: 10.3s

Precision: binary64

Cost: 7104

\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

↓

\[\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\]

\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)

↓

\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))

↓

(FPCore (F l) :precision binary64 (- (* PI l) (/ (/ (tan (* PI l)) F) F)))

double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan(((double) M_PI) * l));
}

↓

double code(double F, double l) {
	return (((double) M_PI) * l) - ((tan(((double) M_PI) * l) / F) / F);
}

Error

The X axis uses an exponential scale — Bits error versus `F`

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Out

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Alternatives

Alternative 1
Error	12.9
Cost	71616

\[\pi \cdot \ell - \frac{\frac{\tan \left(\left(\ell \cdot \sqrt{\pi}\right) \cdot \left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt[3]{\sqrt{\pi}} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)}\right)\right)}{F}}{F}\]

Alternative 2
Error	13.0
Cost	71616

\[\pi \cdot \ell - \frac{\frac{\tan \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\ell \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right)\right)\right)}{F}}{F}\]

Alternative 3
Error	12.8
Cost	59072

\[\pi \cdot \ell - \frac{\frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{\sqrt[3]{F}}}{F}\]

Alternative 4
Error	12.6
Cost	58688

\[\pi \cdot \ell - \frac{\frac{\tan \left(\left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right) \cdot \left(\ell \cdot \left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)\right)}{F}}{F}\]

Alternative 5
Error	13.0
Cost	52288

\[\pi \cdot \ell - \frac{\frac{\tan \left(\sqrt{\pi} \cdot \left(\ell \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right)\right)\right)}{F}}{F}\]

Alternative 6
Error	16.4
Cost	40128

\[\pi \cdot \ell - \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \cdot \left(\sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \cdot \sqrt[3]{\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}}\right)\]

Alternative 7
Error	16.4
Cost	39616

\[\pi \cdot \ell - \frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)} \cdot \left(\sqrt[3]{\tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right)}{F \cdot F}\]

Alternative 8
Error	12.8
Cost	39616

\[\pi \cdot \ell - \frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{F} \cdot \frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{F}\]

Alternative 9
Error	15.2
Cost	39616

\[\pi \cdot \ell - \frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{\frac{F}{\frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{F}}}\]

Alternative 10
Error	12.6
Cost	39360

\[\pi \cdot \ell - \frac{\frac{\tan \left(\left(\ell \cdot \sqrt{\pi}\right) \cdot \left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)}{F}}{F}\]

Alternative 11
Error	39.0
Cost	32960

\[\pi \cdot \ell - \frac{\frac{\tan \left(\left(\sqrt{\pi} \cdot \sqrt{\ell}\right) \cdot \left(\sqrt{\pi} \cdot \sqrt{\ell}\right)\right)}{F}}{F}\]

Alternative 12
Error	40.8
Cost	32960

\[\pi \cdot \ell - \frac{\tan \left(\left(\sqrt{\pi} \cdot \sqrt{\ell}\right) \cdot \left(\sqrt{\pi} \cdot \sqrt{\ell}\right)\right)}{F \cdot F}\]

Alternative 13
Error	12.8
Cost	26816

\[\pi \cdot \ell - \frac{\frac{\tan \left(\sqrt[3]{\pi \cdot \ell} \cdot \left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right)\right)}{F}}{F}\]

Alternative 14
Error	16.9
Cost	26816

\[\sqrt[3]{\pi \cdot \ell} \cdot \left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\]

Alternative 15
Error	29.6
Cost	26816

\[\pi \cdot \ell - \sqrt{\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \cdot \sqrt{\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}}\]

Alternative 16
Error	16.4
Cost	26816

\[\pi \cdot \ell - \frac{\tan \left(\sqrt[3]{\pi \cdot \ell} \cdot \left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right)\right)}{F \cdot F}\]

Alternative 17
Error	12.8
Cost	26688

\[\pi \cdot \ell - \frac{\frac{1}{\sqrt[3]{F} \cdot \sqrt[3]{F}} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{\sqrt[3]{F}}}{F}\]

Alternative 18
Error	12.9
Cost	26560

\[\pi \cdot \ell - \frac{\frac{\tan \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\ell \cdot \sqrt[3]{\pi}\right)\right)}{F}}{F}\]

Alternative 19
Error	16.5
Cost	26560

\[\pi \cdot \ell - \frac{\tan \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\ell \cdot \sqrt[3]{\pi}\right)\right)}{F \cdot F}\]

Alternative 20
Error	40.3
Cost	26560

\[\pi \cdot \ell - \frac{\sqrt{\tan \left(\pi \cdot \ell\right)} \cdot \sqrt{\tan \left(\pi \cdot \ell\right)}}{F \cdot F}\]

Alternative 21
Error	17.1
Cost	26560

\[\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\ell \cdot \sqrt[3]{\pi}\right) - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\]

Alternative 22
Error	39.3
Cost	26560

\[\pi \cdot \ell - \frac{\sqrt{\tan \left(\pi \cdot \ell\right)}}{\frac{F}{\frac{\sqrt{\tan \left(\pi \cdot \ell\right)}}{F}}}\]

Alternative 23
Error	38.5
Cost	26560

\[\pi \cdot \ell - \frac{\sqrt{\tan \left(\pi \cdot \ell\right)}}{F} \cdot \frac{\sqrt{\tan \left(\pi \cdot \ell\right)}}{F}\]

Alternative 24
Error	16.9
Cost	26560

\[\sqrt[3]{\ell} \cdot \left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right) - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\]

Alternative 25
Error	16.4
Cost	26560

\[\pi \cdot \ell - \frac{\tan \left(\sqrt[3]{\ell} \cdot \left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right)\right)}{F \cdot F}\]

Alternative 26
Error	40.8
Cost	20160

\[\sqrt{\pi \cdot \ell} \cdot \sqrt{\pi \cdot \ell} - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\]

Alternative 27
Error	40.7
Cost	20160

\[\pi \cdot \ell - \frac{\tan \left(\sqrt{\pi \cdot \ell} \cdot \sqrt{\pi \cdot \ell}\right)}{F \cdot F}\]

Alternative 28
Error	12.7
Cost	20032

\[\pi \cdot \ell - \frac{\frac{\tan \left(\sqrt{\pi} \cdot \left(\ell \cdot \sqrt{\pi}\right)\right)}{F}}{F}\]

Alternative 29
Error	40.8
Cost	20032

\[\sqrt{\ell} \cdot \left(\pi \cdot \sqrt{\ell}\right) - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\]

Alternative 30
Error	40.7
Cost	20032

\[\pi \cdot \ell - \frac{\tan \left(\sqrt{\ell} \cdot \left(\pi \cdot \sqrt{\ell}\right)\right)}{F \cdot F}\]

Alternative 31
Error	16.4
Cost	20032

\[\pi \cdot \ell - \frac{\tan \left(\sqrt{\pi} \cdot \left(\ell \cdot \sqrt{\pi}\right)\right)}{F \cdot F}\]

Alternative 32
Error	21.8
Cost	19968

\[\pi \cdot \ell - \frac{\sqrt[3]{{\tan \left(\pi \cdot \ell\right)}^{3}}}{F \cdot F}\]

Alternative 33
Error	17.0
Cost	19968

\[\pi \cdot \ell - \frac{\sqrt[3]{{\left(\frac{\tan \left(\pi \cdot \ell\right)}{F}\right)}^{3}}}{F}\]

Alternative 34
Error	42.7
Cost	19968

\[\sqrt[3]{{\left(\pi \cdot \ell\right)}^{3}} - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\]

Alternative 35
Error	29.3
Cost	19904

\[\pi \cdot \ell - \log \left(e^{\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}}\right)\]

Alternative 36
Error	29.9
Cost	19904

\[\pi \cdot \ell - e^{\log \left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)}\]

Alternative 37
Error	40.6
Cost	19904

\[\pi \cdot \ell - \frac{e^{\log \tan \left(\pi \cdot \ell\right)}}{F \cdot F}\]

Alternative 38
Error	41.1
Cost	19904

\[\pi \cdot \ell - \frac{\tan \left(e^{\log \left(\pi \cdot \ell\right)}\right)}{F \cdot F}\]

Alternative 39
Error	24.6
Cost	19904

\[\pi \cdot \ell - \frac{\log \left(e^{\tan \left(\pi \cdot \ell\right)}\right)}{F \cdot F}\]

Alternative 40
Error	16.3
Cost	13760

\[\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\left(F \cdot F\right) \cdot \cos \left(\pi \cdot \ell\right)}\]

Alternative 41
Error	16.3
Cost	13760

\[\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}{F \cdot F}\]

Alternative 42
Error	12.6
Cost	7232

\[\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F}}{F}\]

Alternative 43
Error	12.6
Cost	7232

\[\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}}{F}\]

Alternative 44
Error	12.6
Cost	7232

\[\pi \cdot \ell - \frac{1}{\frac{F}{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}\]

Alternative 45
Error	12.6
Cost	7232

\[\pi \cdot \ell - \frac{1}{F} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{F}\]

Alternative 46
Error	16.5
Cost	7232

\[\pi \cdot \ell - \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}\]

Alternative 47
Error	16.3
Cost	7104

\[\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\]

Alternative 48
Error	21.3
Cost	704

\[\pi \cdot \ell - \frac{\pi \cdot \ell}{F \cdot F}\]

Alternative 49
Error	17.6
Cost	704

\[\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\]

Alternative 50
Error	61.9
Cost	64

\[1\]

Alternative 51
Error	61.9
Cost	64

\[0\]

Alternative 52
Error	61.8
Cost	64

\[-1\]

Error

Derivation

Initial program 16.5
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
Simplified16.3
\[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}}\]
Using strategy rm
Applied associate-/r*_binary6412.6
\[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}}\]
Simplified12.6
\[\leadsto \color{blue}{\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}}\]
Final simplification12.6
\[\leadsto \pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\]

Reproduce

herbie shell --seed 2021022 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))