2tan (problem 3.3.2)

Average Accuracy: 41.6% → 99.0%

Time: 24.6s

Precision: binary64

Cost: 45960

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} t_0 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x} + \tan \varepsilon, \frac{1}{1 - \tan \varepsilon \cdot \tan x}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \sin x \cdot \left(\sin x \cdot {\cos x}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan \varepsilon + \tan x, \frac{1}{1 - \sin x \cdot \frac{\tan \varepsilon}{\cos x}}, t_0\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan x))))
   (if (<= eps -3.3e-9)
     (fma
      (+ (/ (sin x) (cos x)) (tan eps))
      (/ 1.0 (- 1.0 (* (tan eps) (tan x))))
      t_0)
     (if (<= eps 7e-19)
       (* eps (+ 1.0 (* (sin x) (* (sin x) (pow (cos x) -2.0)))))
       (fma
        (+ (tan eps) (tan x))
        (/ 1.0 (- 1.0 (* (sin x) (/ (tan eps) (cos x)))))
        t_0)))))

C

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

↓

double code(double x, double eps) {
	double t_0 = -tan(x);
	double tmp;
	if (eps <= -3.3e-9) {
		tmp = fma(((sin(x) / cos(x)) + tan(eps)), (1.0 / (1.0 - (tan(eps) * tan(x)))), t_0);
	} else if (eps <= 7e-19) {
		tmp = eps * (1.0 + (sin(x) * (sin(x) * pow(cos(x), -2.0))));
	} else {
		tmp = fma((tan(eps) + tan(x)), (1.0 / (1.0 - (sin(x) * (tan(eps) / cos(x))))), t_0);
	}
	return tmp;
}

Julia

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

↓

function code(x, eps)
	t_0 = Float64(-tan(x))
	tmp = 0.0
	if (eps <= -3.3e-9)
		tmp = fma(Float64(Float64(sin(x) / cos(x)) + tan(eps)), Float64(1.0 / Float64(1.0 - Float64(tan(eps) * tan(x)))), t_0);
	elseif (eps <= 7e-19)
		tmp = Float64(eps * Float64(1.0 + Float64(sin(x) * Float64(sin(x) * (cos(x) ^ -2.0)))));
	else
		tmp = fma(Float64(tan(eps) + tan(x)), Float64(1.0 / Float64(1.0 - Float64(sin(x) * Float64(tan(eps) / cos(x))))), t_0);
	end
	return tmp
end

Wolfram

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

↓

code[x_, eps_] := Block[{t$95$0 = (-N[Tan[x], $MachinePrecision])}, If[LessEqual[eps, -3.3e-9], N[(N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[eps, 7e-19], N[(eps * N[(1.0 + N[(N[Sin[x], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * N[Power[N[Cos[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(N[Sin[x], $MachinePrecision] * N[(N[Tan[eps], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Target

Original	41.6%
Target	75.8%
Herbie	99.0%

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

Derivation

1. Split input into 3 regimes

2. if eps < -3.30000000000000018e-9

   1. Initial program 61.7%

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

   2. Applied egg-rr 99.6%

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

      [Start] 61.7	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.5	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 99.6	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      fma-neg [=>] 99.6	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

   3. Applied egg-rr 99.5%

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

      [Start] 99.6	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      tan-quot [=>] 99.6	\[ \mathsf{fma}\left(\color{blue}{\frac{\sin x}{\cos x}} + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      div-inv [=>] 99.6	\[ \mathsf{fma}\left(\color{blue}{\sin x \cdot \frac{1}{\cos x}} + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      fma-def [=>] 99.5	\[ \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sin x, \frac{1}{\cos x}, \tan \varepsilon\right)}, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]

   4. Simplified 99.6%

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

      [Start] 99.5	\[ \mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{1}{\cos x}, \tan \varepsilon\right), \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      fma-udef [=>] 99.6	\[ \mathsf{fma}\left(\color{blue}{\sin x \cdot \frac{1}{\cos x} + \tan \varepsilon}, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      associate-*r/ [=>] 99.6	\[ \mathsf{fma}\left(\color{blue}{\frac{\sin x \cdot 1}{\cos x}} + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      *-rgt-identity [=>] 99.6	\[ \mathsf{fma}\left(\frac{\color{blue}{\sin x}}{\cos x} + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]

   if -3.30000000000000018e-9 < eps < 7.00000000000000031e-19

   1. Initial program 32.5%

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

   2. Taylor expanded in eps around 0 99.7%

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

   3. Simplified 99.7%

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

      [Start] 99.7	\[ \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
      mul-1-neg [=>] 99.7	\[ \varepsilon \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right) \]

   4. Applied egg-rr 99.8%

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

      [Start] 99.7	\[ \varepsilon \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]
      div-inv [=>] 99.7	\[ \varepsilon \cdot \left(1 - \left(-\color{blue}{{\sin x}^{2} \cdot \frac{1}{{\cos x}^{2}}}\right)\right) \]
      unpow2 [=>] 99.7	\[ \varepsilon \cdot \left(1 - \left(-\color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \frac{1}{{\cos x}^{2}}\right)\right) \]
      associate-*l* [=>] 99.8	\[ \varepsilon \cdot \left(1 - \left(-\color{blue}{\sin x \cdot \left(\sin x \cdot \frac{1}{{\cos x}^{2}}\right)}\right)\right) \]
      pow-flip [=>] 99.8	\[ \varepsilon \cdot \left(1 - \left(-\sin x \cdot \left(\sin x \cdot \color{blue}{{\cos x}^{\left(-2\right)}}\right)\right)\right) \]
      metadata-eval [=>] 99.8	\[ \varepsilon \cdot \left(1 - \left(-\sin x \cdot \left(\sin x \cdot {\cos x}^{\color{blue}{-2}}\right)\right)\right) \]

   if 7.00000000000000031e-19 < eps

   1. Initial program 51.0%

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

   2. Applied egg-rr 99.5%

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

      [Start] 51.0	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.5	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 99.5	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      fma-neg [=>] 99.5	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

   3. Applied egg-rr 99.5%

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

      [Start] 99.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      *-commutative [=>] 99.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \color{blue}{\tan \varepsilon \cdot \tan x}}, -\tan x\right) \]
      tan-quot [=>] 99.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan \varepsilon \cdot \color{blue}{\frac{\sin x}{\cos x}}}, -\tan x\right) \]
      associate-*r/ [=>] 99.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \color{blue}{\frac{\tan \varepsilon \cdot \sin x}{\cos x}}}, -\tan x\right) \]

   4. Simplified 99.6%

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

      [Start] 99.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}}, -\tan x\right) \]
      associate-/l* [=>] 99.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \color{blue}{\frac{\tan \varepsilon}{\frac{\cos x}{\sin x}}}}, -\tan x\right) \]
      associate-/r/ [=>] 99.6	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \color{blue}{\frac{\tan \varepsilon}{\cos x} \cdot \sin x}}, -\tan x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin x}{\cos x} + \tan \varepsilon, \frac{1}{1 - \tan \varepsilon \cdot \tan x}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \sin x \cdot \left(\sin x \cdot {\cos x}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan \varepsilon + \tan x, \frac{1}{1 - \sin x \cdot \frac{\tan \varepsilon}{\cos x}}, -\tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	65472

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \frac{t_0}{1 - t_0 \cdot \frac{\sin x}{\cos x}} + \tan \varepsilon \cdot \frac{\tan x}{\frac{1}{\tan x} - \tan \varepsilon} \end{array} \]

Alternative 2
Accuracy	99.0%
Cost	45960

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \sin x \cdot \left(\sin x \cdot {\cos x}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \sin x \cdot \frac{\tan \varepsilon}{\cos x}}, t_0\right)\\ \end{array} \]

Alternative 3
Accuracy	99.1%
Cost	39428

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \sin x \cdot \left(\sin x \cdot {\cos x}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 4
Accuracy	99.1%
Cost	39300

\[\begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \sin x \cdot \left(\sin x \cdot {\cos x}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \]

Alternative 5
Accuracy	99.1%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 7 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \sin x \cdot \left(\sin x \cdot {\cos x}^{-2}\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	99.1%
Cost	32968

\[\begin{array}{l} t_0 := 1 - \tan \varepsilon \cdot \tan x\\ t_1 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_1 - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{-19}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \sin x \cdot \left(\sin x \cdot {\cos x}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0} - \tan x\\ \end{array} \]

Alternative 7
Accuracy	76.8%
Cost	26505

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 7 \cdot 10^{-19}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan \varepsilon + \tan x, 1, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \sin x \cdot \left(\sin x \cdot {\cos x}^{-2}\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	76.9%
Cost	26249

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.4 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 7 \cdot 10^{-19}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan \varepsilon + \tan x, 1, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 9
Accuracy	76.8%
Cost	13449

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.65 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 7 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 10
Accuracy	57.7%
Cost	12992

\[\frac{\sin \varepsilon}{\cos \varepsilon} \]

Alternative 11
Accuracy	56.4%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 7 \cdot 10^{-19}\right):\\ \;\;\;\;\tan \left(\varepsilon + x\right) + \frac{-1}{x \cdot -0.3333333333333333 + \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon\\ \end{array} \]

Alternative 12
Accuracy	54.4%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.55 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 7 \cdot 10^{-19}\right):\\ \;\;\;\;\tan \left(\varepsilon + x\right) - x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon\\ \end{array} \]

Alternative 13
Accuracy	31.3%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023159 
(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)))