2tan (problem 3.3.2)

Average Accuracy: 42.1% → 99.5%

Time: 22.3s

Precision: binary64

Cost: 65544

Math

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

↓

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \frac{\sin x}{\cos x}\\ t_2 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -8.6 \cdot 10^{-8}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_2} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(t_1 + {t_1}^{3}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_2} - \tan x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps)))
        (t_1 (/ (sin x) (cos x)))
        (t_2 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -8.6e-8)
     (- (* t_0 (/ 1.0 t_2)) (tan x))
     (if (<= eps 1.6e-7)
       (fma
        eps
        (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (* (+ t_1 (pow t_1 3.0)) (* eps eps)))
       (- (/ t_0 t_2) (tan x))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = sin(x) / cos(x);
	double t_2 = 1.0 - (tan(x) * tan(eps));
	double tmp;
	if (eps <= -8.6e-8) {
		tmp = (t_0 * (1.0 / t_2)) - tan(x);
	} else if (eps <= 1.6e-7) {
		tmp = fma(eps, (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0))), ((t_1 + pow(t_1, 3.0)) * (eps * eps)));
	} else {
		tmp = (t_0 / t_2) - tan(x);
	}
	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) + tan(eps))
	t_1 = Float64(sin(x) / cos(x))
	t_2 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	tmp = 0.0
	if (eps <= -8.6e-8)
		tmp = Float64(Float64(t_0 * Float64(1.0 / t_2)) - tan(x));
	elseif (eps <= 1.6e-7)
		tmp = fma(eps, Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))), Float64(Float64(t_1 + (t_1 ^ 3.0)) * Float64(eps * eps)));
	else
		tmp = Float64(Float64(t_0 / t_2) - 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_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -8.6e-8], N[(N[(t$95$0 * N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.6e-7], N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / t$95$2), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	42.1%
Target	77.4%
Herbie	99.5%

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

Derivation

1. Split input into 3 regimes

2. if eps < -8.6000000000000002e-8

   1. Initial program 54.9%

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

   2. Applied egg-rr 99.3%

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

      [Start] 54.9	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.3	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 99.3	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   if -8.6000000000000002e-8 < eps < 1.6e-7

   1. Initial program 29.9%

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

   2. Applied egg-rr 30.5%

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

      [Start] 29.9	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 30.5	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 30.5	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      fma-neg [=>] 30.5	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

   3. Applied egg-rr 30.5%

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

      [Start] 30.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
      tan-quot [=>] 30.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}}, -\tan x\right) \]
      associate-*r/ [=>] 30.5	\[ \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \color{blue}{\frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}}, -\tan x\right) \]

   4. Taylor expanded in eps around 0 99.6%

      \[\leadsto \color{blue}{\left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)} \]
      Proof

      [Start] 99.6	\[ \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
      +-commutative [=>] 99.6	\[ \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \cdot {\varepsilon}^{2}} \]
      fma-def [=>] 99.6	\[ \color{blue}{\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \cdot {\varepsilon}^{2}\right)} \]

   if 1.6e-7 < eps

   1. Initial program 53.8%

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

   2. Applied egg-rr 99.3%

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

      [Start] 53.8	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.4	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 99.3	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   3. Simplified 99.4%

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

      [Start] 99.3	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 99.4	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 99.4	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.6 \cdot 10^{-8}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	78336

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \frac{t_0}{1 - t_0 \cdot \frac{\sin x}{\cos x}} + \frac{\tan x \cdot \tan \varepsilon}{\frac{\mathsf{fma}\left(\tan x, -\tan \varepsilon, 1\right)}{\tan x}} \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	59144

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ t_2 := \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(t_2 + {t_2}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.75 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\\ \end{array} \]

Alternative 4
Accuracy	99.4%
Cost	32968

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \]

Alternative 5
Accuracy	78.2%
Cost	32712

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.12 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} - \tan x\\ \end{array} \]

Alternative 6
Accuracy	78.2%
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.12 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} - \tan x\\ \end{array} \]

Alternative 7
Accuracy	78.2%
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.12 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} - \tan x\\ \end{array} \]

Alternative 8
Accuracy	58.4%
Cost	12992

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

Alternative 9
Accuracy	55.6%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.12 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 1.6 \cdot 10^{-7}\right):\\ \;\;\;\;\tan \left(\varepsilon + x\right) - x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon\\ \end{array} \]

Alternative 10
Accuracy	35.7%
Cost	6464

\[\sin \varepsilon \]

Alternative 11
Accuracy	4.3%
Cost	64

\[0 \]

Alternative 12
Accuracy	32.0%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023151 
(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)))