2tan (problem 3.3.2)

Average Error: 57.8% → 0.58%

Time: 20.0s

Precision: binary64

Cost: 72136

Math

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

↓

\[\begin{array}{l} t_0 := 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.95 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, t_0, \frac{\varepsilon \cdot \varepsilon}{\frac{\frac{\cos x}{\sin x}}{t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -1.95e-7)
     (- (/ t_1 (- 1.0 (* (tan x) (tan eps)))) (tan x))
     (if (<= eps 6.2e-10)
       (fma eps t_0 (/ (* eps eps) (/ (/ (cos x) (sin x)) t_0)))
       (- (/ t_1 (- 1.0 (/ (tan x) (/ 1.0 (tan eps))))) (tan x))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = 1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0));
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -1.95e-7) {
		tmp = (t_1 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 6.2e-10) {
		tmp = fma(eps, t_0, ((eps * eps) / ((cos(x) / sin(x)) / t_0)));
	} else {
		tmp = (t_1 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - 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(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))
	t_1 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -1.95e-7)
		tmp = Float64(Float64(t_1 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	elseif (eps <= 6.2e-10)
		tmp = fma(eps, t_0, Float64(Float64(eps * eps) / Float64(Float64(cos(x) / sin(x)) / t_0)));
	else
		tmp = Float64(Float64(t_1 / Float64(1.0 - Float64(tan(x) / Float64(1.0 / tan(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_] := Block[{t$95$0 = N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -1.95e-7], N[(N[(t$95$1 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 6.2e-10], N[(eps * t$95$0 + N[(N[(eps * eps), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	57.8%
Target	23.62%
Herbie	0.58%

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.95000000000000012e-7

   1. Initial program 46.53

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

   2. Applied egg-rr 0.63

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

   3. Simplified 0.61

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

      [Start] 0.63	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 0.61	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 0.61	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   if -1.95000000000000012e-7 < eps < 6.2000000000000003e-10

   1. Initial program 69.6

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

   2. Applied egg-rr 94.91

      \[\leadsto \color{blue}{\left(\tan \left(x + \varepsilon\right) - e^{\mathsf{log1p}\left(\tan x\right)}\right) + 1} \]

   3. Simplified 73.46

      \[\leadsto \color{blue}{\tan \left(\varepsilon + x\right) - \left(e^{\mathsf{log1p}\left(\tan x\right)} - 1\right)} \]
      Proof

      [Start] 94.91	\[ \left(\tan \left(x + \varepsilon\right) - e^{\mathsf{log1p}\left(\tan x\right)}\right) + 1 \]
      associate-+l- [=>] 73.46	\[ \color{blue}{\tan \left(x + \varepsilon\right) - \left(e^{\mathsf{log1p}\left(\tan x\right)} - 1\right)} \]
      +-commutative [=>] 73.46	\[ \tan \color{blue}{\left(\varepsilon + x\right)} - \left(e^{\mathsf{log1p}\left(\tan x\right)} - 1\right) \]

   4. Applied egg-rr 72.12

      \[\leadsto \tan \left(\varepsilon + x\right) - \color{blue}{\sqrt{{\tan x}^{2}}} \]

   5. Taylor expanded in eps around 0 0.42

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

   6. Simplified 0.42

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

      [Start] 0.42	\[ \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} \]
      fma-def [=>] 0.42	\[ \color{blue}{\mathsf{fma}\left(\varepsilon, 1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)} \]
      cancel-sign-sub-inv [=>] 0.42	\[ \mathsf{fma}\left(\varepsilon, \color{blue}{1 + \left(--1\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}}, \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) \]
      metadata-eval [=>] 0.42	\[ \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{1} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) \]
      *-lft-identity [=>] 0.42	\[ \mathsf{fma}\left(\varepsilon, 1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}, \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) \]
      associate-/l* [=>] 0.42	\[ \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \color{blue}{\frac{{\varepsilon}^{2}}{\frac{\cos x}{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}}}\right) \]
      unpow2 [=>] 0.42	\[ \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{\color{blue}{\varepsilon \cdot \varepsilon}}{\frac{\cos x}{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}}\right) \]
      associate-/r* [=>] 0.42	\[ \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{\varepsilon \cdot \varepsilon}{\color{blue}{\frac{\frac{\cos x}{\sin x}}{1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}}}}\right) \]
      cancel-sign-sub-inv [=>] 0.42	\[ \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{\varepsilon \cdot \varepsilon}{\frac{\frac{\cos x}{\sin x}}{\color{blue}{1 + \left(--1\right) \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}}}}\right) \]
      metadata-eval [=>] 0.42	\[ \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{\varepsilon \cdot \varepsilon}{\frac{\frac{\cos x}{\sin x}}{1 + \color{blue}{1} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) \]
      *-lft-identity [=>] 0.42	\[ \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{\varepsilon \cdot \varepsilon}{\frac{\frac{\cos x}{\sin x}}{1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}}}\right) \]

   if 6.2000000000000003e-10 < eps

   1. Initial program 46.51

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

   2. Applied egg-rr 0.88

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

   3. Simplified 0.83

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

      [Start] 0.88	\[ \left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]
      associate-*r/ [=>] 0.83	\[ \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot 1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      *-rgt-identity [=>] 0.83	\[ \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

   4. Applied egg-rr 0.85

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\tan x}{\frac{1}{\tan \varepsilon}}}} - \tan x \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.58

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.95 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{\varepsilon \cdot \varepsilon}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.58%
Cost	72136

\[\begin{array}{l} t_0 := 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, t_0, \frac{\varepsilon}{\frac{\frac{\frac{\cos x}{\sin x}}{t_0}}{\varepsilon}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \end{array} \]

Alternative 2
Error	0.58%
Cost	65736

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \end{array} \]

Alternative 3
Error	0.68%
Cost	33096

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \end{array} \]

Alternative 4
Error	0.68%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.1 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 6.2 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 5
Error	22.2%
Cost	26953

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 6.2 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\varepsilon} + \varepsilon \cdot -0.3333333333333333}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 6
Error	22.61%
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00068:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.38:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 7
Error	22.56%
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00068:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.38:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 8
Error	42.17%
Cost	6464

\[\tan \varepsilon \]

Alternative 9
Error	68.86%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023089 
(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)))