2tan (problem 3.3.2)

Average Accuracy: 42.5% → 99.4%

Time: 23.5s

Precision: binary64

Cost: 52680

Math

\[\tan \left(x + \varepsilon\right) - \tan x \]

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(-\tan \varepsilon\right) - \tan x}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \left(\frac{\tan x}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\right)\\ \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.75e-9)
   (- (/ (- (- (tan eps)) (tan x)) (fma (tan x) (tan eps) -1.0)) (tan x))
   (if (<= eps 9.5e-12)
     (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
     (+
      (/ (tan eps) (- 1.0 (* (tan eps) (tan x))))
      (- (/ (tan x) (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))) (tan x))))))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

↓

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.75e-9) {
		tmp = ((-tan(eps) - tan(x)) / fma(tan(x), tan(eps), -1.0)) - tan(x);
	} else if (eps <= 9.5e-12) {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = (tan(eps) / (1.0 - (tan(eps) * tan(x)))) + ((tan(x) / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

↓

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.75e-9)
		tmp = Float64(Float64(Float64(Float64(-tan(eps)) - tan(x)) / fma(tan(x), tan(eps), -1.0)) - tan(x));
	elseif (eps <= 9.5e-12)
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(tan(eps) / Float64(1.0 - Float64(tan(eps) * tan(x)))) + Float64(Float64(tan(x) / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps)))) - tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := If[LessEqual[eps, -1.75e-9], N[(N[(N[((-N[Tan[eps], $MachinePrecision]) - N[Tan[x], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 9.5e-12], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Tan[eps], $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Tan[x], $MachinePrecision] / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	42.5%
Target	76.1%
Herbie	99.4%

\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

Derivation

1. Split input into 3 regimes

2. if eps < -1.75e-9

   1. Initial program 53.1%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan x + \tan \varepsilon\right)} - \tan x \]
      Proof

      [Start] 53.1	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.3	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 99.3	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-commutative [=>] 99.3	\[ \color{blue}{\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan x + \tan \varepsilon\right)} - \tan x \]

   3. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{-1}{\frac{-1 + \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}}} - \tan x \]
      Proof

      [Start] 99.3	\[ \frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan x + \tan \varepsilon\right) - \tan x \]
      frac-2neg [=>] 99.3	\[ \color{blue}{\frac{-1}{-\left(1 - \tan x \cdot \tan \varepsilon\right)}} \cdot \left(\tan x + \tan \varepsilon\right) - \tan x \]
      associate-*l/ [=>] 99.3	\[ \color{blue}{\frac{\left(-1\right) \cdot \left(\tan x + \tan \varepsilon\right)}{-\left(1 - \tan x \cdot \tan \varepsilon\right)}} - \tan x \]
      associate-/l* [=>] 99.2	\[ \color{blue}{\frac{-1}{\frac{-\left(1 - \tan x \cdot \tan \varepsilon\right)}{\tan x + \tan \varepsilon}}} - \tan x \]
      metadata-eval [=>] 99.2	\[ \frac{\color{blue}{-1}}{\frac{-\left(1 - \tan x \cdot \tan \varepsilon\right)}{\tan x + \tan \varepsilon}} - \tan x \]
      neg-sub0 [=>] 99.2	\[ \frac{-1}{\frac{\color{blue}{0 - \left(1 - \tan x \cdot \tan \varepsilon\right)}}{\tan x + \tan \varepsilon}} - \tan x \]
      metadata-eval [<=] 99.2	\[ \frac{-1}{\frac{\color{blue}{\log 1} - \left(1 - \tan x \cdot \tan \varepsilon\right)}{\tan x + \tan \varepsilon}} - \tan x \]
      associate--r- [=>] 99.2	\[ \frac{-1}{\frac{\color{blue}{\left(\log 1 - 1\right) + \tan x \cdot \tan \varepsilon}}{\tan x + \tan \varepsilon}} - \tan x \]
      metadata-eval [=>] 99.2	\[ \frac{-1}{\frac{\left(\color{blue}{0} - 1\right) + \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x \]
      metadata-eval [=>] 99.2	\[ \frac{-1}{\frac{\color{blue}{-1} + \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x \]

   4. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{\left(-\tan \varepsilon\right) - \tan x}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}} - \tan x \]
      Proof

      [Start] 99.2	\[ \frac{-1}{\frac{-1 + \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x \]
      associate-/l* [<=] 99.3	\[ \color{blue}{\frac{-1 \cdot \left(\tan x + \tan \varepsilon\right)}{-1 + \tan x \cdot \tan \varepsilon}} - \tan x \]
      mul-1-neg [=>] 99.3	\[ \frac{\color{blue}{-\left(\tan x + \tan \varepsilon\right)}}{-1 + \tan x \cdot \tan \varepsilon} - \tan x \]
      neg-sub0 [=>] 99.3	\[ \frac{\color{blue}{0 - \left(\tan x + \tan \varepsilon\right)}}{-1 + \tan x \cdot \tan \varepsilon} - \tan x \]
      +-commutative [=>] 99.3	\[ \frac{0 - \color{blue}{\left(\tan \varepsilon + \tan x\right)}}{-1 + \tan x \cdot \tan \varepsilon} - \tan x \]
      associate--r+ [=>] 99.3	\[ \frac{\color{blue}{\left(0 - \tan \varepsilon\right) - \tan x}}{-1 + \tan x \cdot \tan \varepsilon} - \tan x \]
      neg-sub0 [<=] 99.3	\[ \frac{\color{blue}{\left(-\tan \varepsilon\right)} - \tan x}{-1 + \tan x \cdot \tan \varepsilon} - \tan x \]
      +-commutative [=>] 99.3	\[ \frac{\left(-\tan \varepsilon\right) - \tan x}{\color{blue}{\tan x \cdot \tan \varepsilon + -1}} - \tan x \]
      fma-def [=>] 99.3	\[ \frac{\left(-\tan \varepsilon\right) - \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}} - \tan x \]

   if -1.75e-9 < eps < 9.4999999999999995e-12

   1. Initial program 30.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in eps around 0 99.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}} \]
      Proof

      [Start] 99.5	\[ \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
      cancel-sign-sub-inv [=>] 99.5	\[ \varepsilon \cdot \color{blue}{\left(1 + \left(--1\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
      distribute-lft-in [=>] 99.6	\[ \color{blue}{\varepsilon \cdot 1 + \varepsilon \cdot \left(\left(--1\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
      *-commutative [<=] 99.6	\[ \color{blue}{1 \cdot \varepsilon} + \varepsilon \cdot \left(\left(--1\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
      *-lft-identity [=>] 99.6	\[ \color{blue}{\varepsilon} + \varepsilon \cdot \left(\left(--1\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
      distribute-lft-neg-in [<=] 99.6	\[ \varepsilon + \varepsilon \cdot \color{blue}{\left(--1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
      mul-1-neg [=>] 99.6	\[ \varepsilon + \varepsilon \cdot \left(-\color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right) \]
      remove-double-neg [=>] 99.6	\[ \varepsilon + \varepsilon \cdot \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}} \]

   if 9.4999999999999995e-12 < eps

   1. Initial program 53.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan x + \tan \varepsilon\right)} - \tan x \]
      Proof

      [Start] 53.5	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.1	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 99.1	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-commutative [=>] 99.1	\[ \color{blue}{\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan x + \tan \varepsilon\right)} - \tan x \]

   3. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right)} \]
      Proof

      [Start] 99.1	\[ \frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \left(\tan x + \tan \varepsilon\right) - \tan x \]
      distribute-lft-in [=>] 99.1	\[ \color{blue}{\left(\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \tan x + \frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \tan \varepsilon\right)} - \tan x \]
      +-commutative [=>] 99.1	\[ \color{blue}{\left(\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \tan \varepsilon + \frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \tan x\right)} - \tan x \]
      associate--l+ [=>] 99.1	\[ \color{blue}{\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \tan \varepsilon + \left(\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \tan x - \tan x\right)} \]
      *-commutative [=>] 99.1	\[ \color{blue}{\tan \varepsilon \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} + \left(\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \tan x - \tan x\right) \]
      un-div-inv [=>] 99.1	\[ \color{blue}{\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} + \left(\frac{1}{1 - \tan x \cdot \tan \varepsilon} \cdot \tan x - \tan x\right) \]
      *-commutative [=>] 99.1	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\color{blue}{\tan x \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\right) \]
      un-div-inv [=>] 99.1	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\color{blue}{\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\right) \]

   4. Applied egg-rr 99.1%

      \[\leadsto \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan x}{1 - \color{blue}{\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}} - \tan x\right) \]
      Proof

      [Start] 99.1	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan x}{1 - \tan x \cdot \tan \varepsilon} - \tan x\right) \]
      tan-quot [=>] 99.1	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan x}{1 - \tan x \cdot \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}} - \tan x\right) \]
      associate-*r/ [=>] 99.1	\[ \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \left(\frac{\tan x}{1 - \color{blue}{\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}} - \tan x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(-\tan \varepsilon\right) - \tan x}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon}{1 - \tan \varepsilon \cdot \tan x} + \left(\frac{\tan x}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	39305

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 9.5 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\left(-\tan \varepsilon\right) - \tan x}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	33096

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \tan x\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 9.5 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 4
Accuracy	77.1%
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 5
Accuracy	77.0%
Cost	26376

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{expm1}\left(\varepsilon + \varepsilon \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 6
Accuracy	58.0%
Cost	6464

\[\tan \varepsilon \]

Alternative 7
Accuracy	4.2%
Cost	64

\[0 \]

Alternative 8
Accuracy	30.9%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023133 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))