2tan (problem 3.3.2)

Average Error: 37.5 → 14.0

Time: 21.1s

Precision: binary64

Cost: 347080

Math

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

↓

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{2}\\ t_2 := \frac{t_0}{t_1}\\ t_3 := 1 + t_2\\ t_4 := 0.16666666666666666 + \left(0.16666666666666666 \cdot t_2 + \left(\left(-\frac{t_0 \cdot t_3}{t_1}\right) + -0.5 \cdot t_3\right)\right)\\ \mathbf{if}\;\varepsilon \leq -0.0038:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_3 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right)}{\cos x} + \left(\varepsilon \cdot t_3 + \left(-\left({\varepsilon}^{3} \cdot t_4 + \left(\frac{\sin x \cdot t_4}{\cos x} + \frac{\sin x \cdot t_3}{\cos x} \cdot -0.3333333333333333\right) \cdot {\varepsilon}^{4}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0))
        (t_1 (pow (cos x) 2.0))
        (t_2 (/ t_0 t_1))
        (t_3 (+ 1.0 t_2))
        (t_4
         (+
          0.16666666666666666
          (+
           (* 0.16666666666666666 t_2)
           (+ (- (/ (* t_0 t_3) t_1)) (* -0.5 t_3))))))
   (if (<= eps -0.0038)
     (tan eps)
     (if (<= eps 6.5e-5)
       (+
        (/ (* t_3 (* (sin x) (pow eps 2.0))) (cos x))
        (+
         (* eps t_3)
         (-
          (+
           (* (pow eps 3.0) t_4)
           (*
            (+
             (/ (* (sin x) t_4) (cos x))
             (* (/ (* (sin x) t_3) (cos x)) -0.3333333333333333))
            (pow eps 4.0))))))
       (tan eps)))))

C

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

↓

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = pow(cos(x), 2.0);
	double t_2 = t_0 / t_1;
	double t_3 = 1.0 + t_2;
	double t_4 = 0.16666666666666666 + ((0.16666666666666666 * t_2) + (-((t_0 * t_3) / t_1) + (-0.5 * t_3)));
	double tmp;
	if (eps <= -0.0038) {
		tmp = tan(eps);
	} else if (eps <= 6.5e-5) {
		tmp = ((t_3 * (sin(x) * pow(eps, 2.0))) / cos(x)) + ((eps * t_3) + -((pow(eps, 3.0) * t_4) + ((((sin(x) * t_4) / cos(x)) + (((sin(x) * t_3) / cos(x)) * -0.3333333333333333)) * pow(eps, 4.0))));
	} else {
		tmp = tan(eps);
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = sin(x) ** 2.0d0
    t_1 = cos(x) ** 2.0d0
    t_2 = t_0 / t_1
    t_3 = 1.0d0 + t_2
    t_4 = 0.16666666666666666d0 + ((0.16666666666666666d0 * t_2) + (-((t_0 * t_3) / t_1) + ((-0.5d0) * t_3)))
    if (eps <= (-0.0038d0)) then
        tmp = tan(eps)
    else if (eps <= 6.5d-5) then
        tmp = ((t_3 * (sin(x) * (eps ** 2.0d0))) / cos(x)) + ((eps * t_3) + -(((eps ** 3.0d0) * t_4) + ((((sin(x) * t_4) / cos(x)) + (((sin(x) * t_3) / cos(x)) * (-0.3333333333333333d0))) * (eps ** 4.0d0))))
    else
        tmp = tan(eps)
    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.pow(Math.sin(x), 2.0);
	double t_1 = Math.pow(Math.cos(x), 2.0);
	double t_2 = t_0 / t_1;
	double t_3 = 1.0 + t_2;
	double t_4 = 0.16666666666666666 + ((0.16666666666666666 * t_2) + (-((t_0 * t_3) / t_1) + (-0.5 * t_3)));
	double tmp;
	if (eps <= -0.0038) {
		tmp = Math.tan(eps);
	} else if (eps <= 6.5e-5) {
		tmp = ((t_3 * (Math.sin(x) * Math.pow(eps, 2.0))) / Math.cos(x)) + ((eps * t_3) + -((Math.pow(eps, 3.0) * t_4) + ((((Math.sin(x) * t_4) / Math.cos(x)) + (((Math.sin(x) * t_3) / Math.cos(x)) * -0.3333333333333333)) * Math.pow(eps, 4.0))));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

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

↓

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = math.pow(math.cos(x), 2.0)
	t_2 = t_0 / t_1
	t_3 = 1.0 + t_2
	t_4 = 0.16666666666666666 + ((0.16666666666666666 * t_2) + (-((t_0 * t_3) / t_1) + (-0.5 * t_3)))
	tmp = 0
	if eps <= -0.0038:
		tmp = math.tan(eps)
	elif eps <= 6.5e-5:
		tmp = ((t_3 * (math.sin(x) * math.pow(eps, 2.0))) / math.cos(x)) + ((eps * t_3) + -((math.pow(eps, 3.0) * t_4) + ((((math.sin(x) * t_4) / math.cos(x)) + (((math.sin(x) * t_3) / math.cos(x)) * -0.3333333333333333)) * math.pow(eps, 4.0))))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

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

↓

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(1.0 + t_2)
	t_4 = Float64(0.16666666666666666 + Float64(Float64(0.16666666666666666 * t_2) + Float64(Float64(-Float64(Float64(t_0 * t_3) / t_1)) + Float64(-0.5 * t_3))))
	tmp = 0.0
	if (eps <= -0.0038)
		tmp = tan(eps);
	elseif (eps <= 6.5e-5)
		tmp = Float64(Float64(Float64(t_3 * Float64(sin(x) * (eps ^ 2.0))) / cos(x)) + Float64(Float64(eps * t_3) + Float64(-Float64(Float64((eps ^ 3.0) * t_4) + Float64(Float64(Float64(Float64(sin(x) * t_4) / cos(x)) + Float64(Float64(Float64(sin(x) * t_3) / cos(x)) * -0.3333333333333333)) * (eps ^ 4.0))))));
	else
		tmp = tan(eps);
	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 = sin(x) ^ 2.0;
	t_1 = cos(x) ^ 2.0;
	t_2 = t_0 / t_1;
	t_3 = 1.0 + t_2;
	t_4 = 0.16666666666666666 + ((0.16666666666666666 * t_2) + (-((t_0 * t_3) / t_1) + (-0.5 * t_3)));
	tmp = 0.0;
	if (eps <= -0.0038)
		tmp = tan(eps);
	elseif (eps <= 6.5e-5)
		tmp = ((t_3 * (sin(x) * (eps ^ 2.0))) / cos(x)) + ((eps * t_3) + -(((eps ^ 3.0) * t_4) + ((((sin(x) * t_4) / cos(x)) + (((sin(x) * t_3) / cos(x)) * -0.3333333333333333)) * (eps ^ 4.0))));
	else
		tmp = tan(eps);
	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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(0.16666666666666666 + N[(N[(0.16666666666666666 * t$95$2), $MachinePrecision] + N[((-N[(N[(t$95$0 * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision]) + N[(-0.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.0038], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 6.5e-5], N[(N[(N[(t$95$3 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[(eps * t$95$3), $MachinePrecision] + (-N[(N[(N[Power[eps, 3.0], $MachinePrecision] * t$95$4), $MachinePrecision] + N[(N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$4), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]]]]]]

Target

Original	37.5
Target	15.0
Herbie	14.0

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

Derivation

1. Split input into 2 regimes

2. if eps < -0.00379999999999999999 or 6.49999999999999943e-5 < eps

   1. Initial program 29.9

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

   2. Taylor expanded in x around 0 28.4

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} - \tan x \]

   3. Simplified 28.3

      \[\leadsto \color{blue}{\tan \varepsilon} - \tan x \]
      Proof

      [Start] 28.4	\[ \frac{\sin \varepsilon}{\cos \varepsilon} - \tan x \]
      trigometric-lifting-simplify-12 [=>] 28.3	\[ \color{blue}{\tan \varepsilon} - \tan x \]

   4. Taylor expanded in x around 0 28.0

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

   5. Simplified 27.8

      \[\leadsto \color{blue}{\tan \varepsilon} \]
      Proof

      [Start] 28.0	\[ \frac{\sin \varepsilon}{\cos \varepsilon} \]
      trigometric-lifting-simplify-11 [<=] 27.8	\[ \color{blue}{\tan \varepsilon} \]

   if -0.00379999999999999999 < eps < 6.49999999999999943e-5

   1. Initial program 45.1

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

   2. Taylor expanded in eps around 0 0.3

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} + \left(-1 \cdot \left(\left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{\left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) \cdot \sin x}{\cos x} + 0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) \cdot {\varepsilon}^{4}\right) + -1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right)\right)} \]

   3. Simplified 0.3

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \left(\sin x \cdot {\varepsilon}^{2}\right)}{\cos x} + \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(-\left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\left(-\frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right) + -0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \left(\frac{\sin x \cdot \left(0.16666666666666666 + \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\left(-\frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right) + -0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)}{\cos x} + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} \cdot -0.3333333333333333\right) \cdot {\varepsilon}^{4}\right)\right)\right)} \]
      Proof

      [Start] 0.3

      \[ \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} + \left(-1 \cdot \left(\left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(\frac{\left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) \cdot \sin x}{\cos x} + 0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) \cdot {\varepsilon}^{4}\right) + -1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + -0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 14.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0038:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \left(\sin x \cdot {\varepsilon}^{2}\right)}{\cos x} + \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(-\left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\left(-\frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right) + -0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \left(\frac{\sin x \cdot \left(0.16666666666666666 + \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\left(-\frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right) + -0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)}{\cos x} + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} \cdot -0.3333333333333333\right) \cdot {\varepsilon}^{4}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternatives

Alternative 1
Error	14.0
Cost	183496

\[\begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{2}\\ t_2 := \frac{t_0}{t_1}\\ t_3 := 1 + t_2\\ \mathbf{if}\;\varepsilon \leq -0.0095:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_3 \cdot \left(\sin x \cdot {\varepsilon}^{2}\right)}{\cos x} + \left(\varepsilon \cdot t_3 + \left(0.16666666666666666 + \left(0.16666666666666666 \cdot t_2 + \left(\left(-\frac{t_0 \cdot t_3}{t_1}\right) + -0.5 \cdot t_3\right)\right)\right) \cdot \left(-{\varepsilon}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 2
Error	14.1
Cost	72200

\[\begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\\ \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot t_0 + \frac{\sin x \cdot \left(t_0 \cdot {\varepsilon}^{2}\right)}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 3
Error	14.3
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.12 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{-6}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 4
Error	41.5
Cost	6464

\[\sin \varepsilon \]

Alternative 5
Error	26.9
Cost	6464

\[\tan \varepsilon \]

Alternative 6
Error	43.9
Cost	64

\[\varepsilon \]

Reproduce

herbie shell --seed 2023092 
(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)))