2tan (problem 3.3.2)

Percentage Accurate: 42.6% → 99.5%

Time: 18.9s

Precision: binary64

Cost: 59144

Math

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

↓

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ t_2 := \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -4.25 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(t_2 + {t_2}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \end{array} \]

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 (- 1.0 (* (tan x) (tan eps))))
        (t_2 (/ (sin x) (cos x))))
   (if (<= eps -4.25e-7)
     (- (/ t_0 t_1) (tan x))
     (if (<= eps 8.2e-7)
       (*
        eps
        (+
         (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
         (* eps (+ t_2 (pow t_2 3.0)))))
       (- (* t_0 (/ 1.0 t_1)) (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 = 1.0 - (tan(x) * tan(eps));
	double t_2 = sin(x) / cos(x);
	double tmp;
	if (eps <= -4.25e-7) {
		tmp = (t_0 / t_1) - tan(x);
	} else if (eps <= 8.2e-7) {
		tmp = eps * ((1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0))) + (eps * (t_2 + pow(t_2, 3.0))));
	} else {
		tmp = (t_0 * (1.0 / t_1)) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = tan(x) + tan(eps)
    t_1 = 1.0d0 - (tan(x) * tan(eps))
    t_2 = sin(x) / cos(x)
    if (eps <= (-4.25d-7)) then
        tmp = (t_0 / t_1) - tan(x)
    else if (eps <= 8.2d-7) then
        tmp = eps * ((1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0))) + (eps * (t_2 + (t_2 ** 3.0d0))))
    else
        tmp = (t_0 * (1.0d0 / t_1)) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

↓

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double t_1 = 1.0 - (Math.tan(x) * Math.tan(eps));
	double t_2 = Math.sin(x) / Math.cos(x);
	double tmp;
	if (eps <= -4.25e-7) {
		tmp = (t_0 / t_1) - Math.tan(x);
	} else if (eps <= 8.2e-7) {
		tmp = eps * ((1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0))) + (eps * (t_2 + Math.pow(t_2, 3.0))));
	} else {
		tmp = (t_0 * (1.0 / t_1)) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

↓

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	t_1 = 1.0 - (math.tan(x) * math.tan(eps))
	t_2 = math.sin(x) / math.cos(x)
	tmp = 0
	if eps <= -4.25e-7:
		tmp = (t_0 / t_1) - math.tan(x)
	elif eps <= 8.2e-7:
		tmp = eps * ((1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0))) + (eps * (t_2 + math.pow(t_2, 3.0))))
	else:
		tmp = (t_0 * (1.0 / t_1)) - math.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(1.0 - Float64(tan(x) * tan(eps)))
	t_2 = Float64(sin(x) / cos(x))
	tmp = 0.0
	if (eps <= -4.25e-7)
		tmp = Float64(Float64(t_0 / t_1) - tan(x));
	elseif (eps <= 8.2e-7)
		tmp = Float64(eps * Float64(Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))) + Float64(eps * Float64(t_2 + (t_2 ^ 3.0)))));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 / t_1)) - tan(x));
	end
	return tmp
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

↓

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	t_1 = 1.0 - (tan(x) * tan(eps));
	t_2 = sin(x) / cos(x);
	tmp = 0.0;
	if (eps <= -4.25e-7)
		tmp = (t_0 / t_1) - tan(x);
	elseif (eps <= 8.2e-7)
		tmp = eps * ((1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0))) + (eps * (t_2 + (t_2 ^ 3.0))));
	else
		tmp = (t_0 * (1.0 / t_1)) - tan(x);
	end
	tmp_2 = 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[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.25e-7], N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 8.2e-7], N[(eps * N[(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[(eps * N[(t$95$2 + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	42.6%
Target	76.1%
Herbie	99.5%

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

Derivation

1. Split input into 3 regimes

2. if eps < -4.25000000000000007e-7

   1. Initial program 51.9%

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

   2. Applied egg-rr 99.6%

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

      [Start] 51.9	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.6	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      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. Simplified 99.6%

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

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

   if -4.25000000000000007e-7 < eps < 8.1999999999999998e-7

   1. Initial program 35.4%

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

   2. Applied egg-rr 37.1%

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

      [Start] 35.4	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 37.1	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 37.1	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      fma-neg [=>] 37.1	\[ \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

   3. Taylor expanded in eps around 0 99.6%

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

   4. Simplified 99.6%

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

      [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	\[ \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \cdot {\varepsilon}^{2}} \]
      *-commutative [=>] 99.6	\[ \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)} \]
      unpow2 [=>] 99.6	\[ \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \]
      associate-*l* [=>] 99.6	\[ \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)} \]
      distribute-lft-out [=>] 99.6	\[ \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)} \]
      cube-mult [=>] 99.6	\[ \varepsilon \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\frac{\sin x}{\cos x} + \frac{\color{blue}{\sin x \cdot \left(\sin x \cdot \sin x\right)}}{{\cos x}^{3}}\right)\right) \]
      unpow2 [<=] 99.6	\[ \varepsilon \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\frac{\sin x}{\cos x} + \frac{\sin x \cdot \color{blue}{{\sin x}^{2}}}{{\cos x}^{3}}\right)\right) \]

   if 8.1999999999999998e-7 < eps

   1. Initial program 57.6%

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

   2. Applied egg-rr 99.5%

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

      [Start] 57.6	\[ \tan \left(x + \varepsilon\right) - \tan x \]
      tan-sum [=>] 99.3	\[ \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
      div-inv [=>] 99.5	\[ \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.25 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	33096

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.3 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \tan x, \tan x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \tan x, \tan x, \varepsilon\right)\\ \end{array} \]

Alternative 3
Accuracy	77.3%
Cost	19784

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 6.6 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \tan x, \tan x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 4
Accuracy	77.3%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.22 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 5
Accuracy	58.3%
Cost	6464

\[\tan \varepsilon \]

Alternative 6
Accuracy	30.8%
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023161 
(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)))