2tan (problem 3.3.2)

Average Error: 36.5 → 0.5

Time: 20.4s

Precision: binary64

Cost: 117896

Math

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

↓

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_2 := -\tan x\\ t_3 := \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -14.371172738720198:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, t_2\right)\\ \mathbf{elif}\;\varepsilon \leq 7.324121520452104 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + t_1, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(t_3 + {t_3}^{3}\right) + \varepsilon \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + t_1 \cdot 1.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)}, t_2\right)\\ \end{array} \]

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 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_2 (- (tan x)))
        (t_3 (/ (sin x) (cos x))))
   (if (<= eps -14.371172738720198)
     (fma t_0 (/ 1.0 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))) t_2)
     (if (<= eps 7.324121520452104e-7)
       (fma
        eps
        (+ 1.0 t_1)
        (*
         (* eps eps)
         (+
          (+ t_3 (pow t_3 3.0))
          (*
           eps
           (+
            0.3333333333333333
            (+
             (/ (pow (sin x) 4.0) (pow (cos x) 4.0))
             (* t_1 1.3333333333333333)))))))
       (fma t_0 (/ 1.0 (- 1.0 (log (exp (* (tan x) (tan eps)))))) t_2)))))

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 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double t_2 = -tan(x);
	double t_3 = sin(x) / cos(x);
	double tmp;
	if (eps <= -14.371172738720198) {
		tmp = fma(t_0, (1.0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))), t_2);
	} else if (eps <= 7.324121520452104e-7) {
		tmp = fma(eps, (1.0 + t_1), ((eps * eps) * ((t_3 + pow(t_3, 3.0)) + (eps * (0.3333333333333333 + ((pow(sin(x), 4.0) / pow(cos(x), 4.0)) + (t_1 * 1.3333333333333333)))))));
	} else {
		tmp = fma(t_0, (1.0 / (1.0 - log(exp((tan(x) * tan(eps)))))), t_2);
	}
	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) ^ 2.0) / (cos(x) ^ 2.0))
	t_2 = Float64(-tan(x))
	t_3 = Float64(sin(x) / cos(x))
	tmp = 0.0
	if (eps <= -14.371172738720198)
		tmp = fma(t_0, Float64(1.0 / Float64(1.0 - Float64(Float64(tan(x) * sin(eps)) / cos(eps)))), t_2);
	elseif (eps <= 7.324121520452104e-7)
		tmp = fma(eps, Float64(1.0 + t_1), Float64(Float64(eps * eps) * Float64(Float64(t_3 + (t_3 ^ 3.0)) + Float64(eps * Float64(0.3333333333333333 + Float64(Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)) + Float64(t_1 * 1.3333333333333333)))))));
	else
		tmp = fma(t_0, Float64(1.0 / Float64(1.0 - log(exp(Float64(tan(x) * tan(eps)))))), t_2);
	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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Tan[x], $MachinePrecision])}, Block[{t$95$3 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -14.371172738720198], N[(t$95$0 * N[(1.0 / N[(1.0 - N[(N[(N[Tan[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[eps, 7.324121520452104e-7], N[(eps * N[(1.0 + t$95$1), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(t$95$3 + N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(0.3333333333333333 + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(1.0 - N[Log[N[Exp[N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]]

Target

Original	36.5
Target	15.1
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if eps < -14.371172738720198

   1. Initial program 29.0

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

   2. Applied egg-rr 0.3

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

   3. Applied egg-rr 0.3

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

   if -14.371172738720198 < eps < 7.3241215204521039e-7

   1. Initial program 43.9

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

   2. Applied egg-rr 43.3

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

   3. Taylor expanded in eps around 0 0.5

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(-1 \cdot \left({\varepsilon}^{2} \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -1 \cdot \frac{\sin x}{\cos x}\right)\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)} \]

   4. Simplified 0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right) + \varepsilon \cdot \left(0.3333333333333333 - \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -1.3333333333333333 - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)\right)} \]

   if 7.3241215204521039e-7 < eps

   1. Initial program 29.8

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

   2. Applied egg-rr 0.4

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

   3. Applied egg-rr 0.4

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

   4. Applied egg-rr 0.5

      \[\leadsto \mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \color{blue}{\log \left(e^{\tan x \cdot \tan \varepsilon}\right)}}, -\tan x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -14.371172738720198:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 7.324121520452104 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right) + \varepsilon \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot 1.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)}, -\tan x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	98376

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_3 := 1 + t_2\\ \mathbf{if}\;\varepsilon \leq -14.371172738720198:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 7.324121520452104 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{log1p}\left(\varepsilon \cdot \left(t_3 + \varepsilon \cdot \left(t_3 \cdot \left(\frac{\sin x}{\cos x} + \left(0.5 + t_2 \cdot 0.5\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)}, t_0\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	65544

\[\begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := -\tan x\\ t_2 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -14.371172738720198:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 7.324121520452104 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon \cdot \left(\varepsilon \cdot \left(t_0 + {t_0}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_2, \frac{1}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)}, t_1\right)\\ \end{array} \]

Alternative 3
Error	0.7
Cost	52360

\[\begin{array}{l} t_0 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -14.371172738720198:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 7.954732301148908 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{0.5 + \cos \left(x \cdot 2\right) \cdot -0.5}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\sin x, \frac{1}{\cos x}, \tan \varepsilon\right), \frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_0\right)\\ \end{array} \]

Alternative 4
Error	0.7
Cost	52232

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -14.371172738720198:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, t_0\right)\\ \mathbf{elif}\;\varepsilon \leq 7.954732301148908 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{0.5 + \cos \left(x \cdot 2\right) \cdot -0.5}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)}, t_0\right)\\ \end{array} \]

Alternative 5
Error	0.7
Cost	45828

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -14.371172738720198:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 7.954732301148908 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{0.5 + \cos \left(x \cdot 2\right) \cdot -0.5}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 6
Error	0.7
Cost	39364

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -14.371172738720198:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7.954732301148908 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{0.5 + \cos \left(x \cdot 2\right) \cdot -0.5}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 7
Error	0.7
Cost	39300

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -14.371172738720198:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 7.954732301148908 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{0.5 + \cos \left(x \cdot 2\right) \cdot -0.5}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \]

Alternative 8
Error	0.6
Cost	39172

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -14.371172738720198:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(\tan \varepsilon, -\tan x, 1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 7.954732301148908 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{0.5 + \cos \left(x \cdot 2\right) \cdot -0.5}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 9
Error	0.7
Cost	32968

\[\begin{array}{l} t_0 := \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{if}\;\varepsilon \leq -14.371172738720198:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.954732301148908 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{0.5 + \cos \left(x \cdot 2\right) \cdot -0.5}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	14.5
Cost	26632

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \mathbf{if}\;\varepsilon \leq -14.371172738720198:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.324121520452104 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{0.5 + \cos \left(x \cdot 2\right) \cdot -0.5}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	14.5
Cost	26248

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -14.371172738720198:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.324121520452104 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\left(\frac{\sin x}{\cos x}\right)}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	14.5
Cost	26248

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \mathbf{if}\;\varepsilon \leq -14.371172738720198:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.324121520452104 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\left(\frac{\sin x}{\cos x}\right)}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	14.6
Cost	19976

\[\begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -14.371172738720198:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 7.324121520452104 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	26.5
Cost	12992

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

Alternative 15
Error	44.1
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2022216 
(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)))