sintan (problem 3.4.5)

Average Error: 31.3 → 0.6

Time: 17.9s

Precision: binary64

Cost: 13512

Math

\[\frac{x - \sin x}{x - \tan x} \]

↓

\[\begin{array}{l} t_0 := \frac{\sin x - x}{\tan x - x}\\ \mathbf{if}\;x \leq -136948893217.82822:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0028624513617995866:\\ \;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot {x}^{4}\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- (sin x) x) (- (tan x) x))))
   (if (<= x -136948893217.82822)
     t_0
     (if (<= x 0.0028624513617995866)
       (+ (+ (* 0.225 (* x x)) (* -0.009642857142857142 (pow x 4.0))) -0.5)
       t_0))))

C

double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}

↓

double code(double x) {
	double t_0 = (sin(x) - x) / (tan(x) - x);
	double tmp;
	if (x <= -136948893217.82822) {
		tmp = t_0;
	} else if (x <= 0.0028624513617995866) {
		tmp = ((0.225 * (x * x)) + (-0.009642857142857142 * pow(x, 4.0))) + -0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x - sin(x)) / (x - tan(x))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(x) - x) / (tan(x) - x)
    if (x <= (-136948893217.82822d0)) then
        tmp = t_0
    else if (x <= 0.0028624513617995866d0) then
        tmp = ((0.225d0 * (x * x)) + ((-0.009642857142857142d0) * (x ** 4.0d0))) + (-0.5d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return (x - Math.sin(x)) / (x - Math.tan(x));
}

↓

public static double code(double x) {
	double t_0 = (Math.sin(x) - x) / (Math.tan(x) - x);
	double tmp;
	if (x <= -136948893217.82822) {
		tmp = t_0;
	} else if (x <= 0.0028624513617995866) {
		tmp = ((0.225 * (x * x)) + (-0.009642857142857142 * Math.pow(x, 4.0))) + -0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	return (x - math.sin(x)) / (x - math.tan(x))

↓

def code(x):
	t_0 = (math.sin(x) - x) / (math.tan(x) - x)
	tmp = 0
	if x <= -136948893217.82822:
		tmp = t_0
	elif x <= 0.0028624513617995866:
		tmp = ((0.225 * (x * x)) + (-0.009642857142857142 * math.pow(x, 4.0))) + -0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	return Float64(Float64(x - sin(x)) / Float64(x - tan(x)))
end

↓

function code(x)
	t_0 = Float64(Float64(sin(x) - x) / Float64(tan(x) - x))
	tmp = 0.0
	if (x <= -136948893217.82822)
		tmp = t_0;
	elseif (x <= 0.0028624513617995866)
		tmp = Float64(Float64(Float64(0.225 * Float64(x * x)) + Float64(-0.009642857142857142 * (x ^ 4.0))) + -0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = (x - sin(x)) / (x - tan(x));
end

↓

function tmp_2 = code(x)
	t_0 = (sin(x) - x) / (tan(x) - x);
	tmp = 0.0;
	if (x <= -136948893217.82822)
		tmp = t_0;
	elseif (x <= 0.0028624513617995866)
		tmp = ((0.225 * (x * x)) + (-0.009642857142857142 * (x ^ 4.0))) + -0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -136948893217.82822], t$95$0, If[LessEqual[x, 0.0028624513617995866], N[(N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -136948893217.82822 or 0.00286245136179958662 < x

   1. Initial program 0.0

      \[\frac{x - \sin x}{x - \tan x} \]

   2. Simplified 0.0

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
      Proof
      (/.f64 (-.f64 (sin.f64 x) x) (-.f64 (tan.f64 x) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 (sin.f64 x) x) (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (-.f64 (sin.f64 x) x) (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 (sin.f64 x) x)) (*.f64 -1 (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (sin.f64 x) x))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sin.f64 x) x))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (sin.f64 x)) x)) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sin.f64 x))) x) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (sin.f64 x)))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x (sin.f64 x))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (tan.f64 x)) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (tan.f64 x))) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (tan.f64 x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= sub-neg_binary64 (-.f64 x (tan.f64 x)))): 0 points increase in error, 0 points decrease in error

   if -136948893217.82822 < x < 0.00286245136179958662

   1. Initial program 62.0

      \[\frac{x - \sin x}{x - \tan x} \]

   2. Simplified 62.0

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
      Proof
      (/.f64 (-.f64 (sin.f64 x) x) (-.f64 (tan.f64 x) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 (sin.f64 x) x) (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (-.f64 (sin.f64 x) x) (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 (sin.f64 x) x)) (*.f64 -1 (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (sin.f64 x) x))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sin.f64 x) x))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (sin.f64 x)) x)) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sin.f64 x))) x) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (sin.f64 x)))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x (sin.f64 x))) (*.f64 -1 (-.f64 (tan.f64 x) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (tan.f64 x) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (tan.f64 x)) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (tan.f64 x))) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (tan.f64 x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 x (sin.f64 x)) (Rewrite<= sub-neg_binary64 (-.f64 x (tan.f64 x)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around 0 1.1

      \[\leadsto \color{blue}{\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5} \]

   4. Applied egg-rr 1.1

      \[\leadsto \left(\color{blue}{\left(0 + 0.225 \cdot \left(x \cdot x\right)\right)} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -136948893217.82822:\\ \;\;\;\;\frac{\sin x - x}{\tan x - x}\\ \mathbf{elif}\;x \leq 0.0028624513617995866:\\ \;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot {x}^{4}\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x - x}{\tan x - x}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	7432

\[\begin{array}{l} t_0 := \frac{x + \tan x}{x}\\ \mathbf{if}\;x \leq -136948893217.82822:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0028624513617995866:\\ \;\;\;\;\left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot {x}^{4}\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.0
Cost	6984

\[\begin{array}{l} t_0 := \frac{x + \tan x}{x}\\ \mathbf{if}\;x \leq -136948893217.82822:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.0028624513617995866:\\ \;\;\;\;1 + \left(x \cdot \left(x \cdot 0.225\right) + -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -136948893217.82822:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.0028624513617995866:\\ \;\;\;\;1 + \left(x \cdot \left(x \cdot 0.225\right) + -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	1.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -136948893217.82822:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.0028624513617995866:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	1.2
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -136948893217.82822:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.0028624513617995866:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	31.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022300 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))