sintan (problem 3.4.5)

Average Error: 31.7 → 0.0

Time: 21.2s

Precision: binary64

Cost: 20488

Math

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

↓

\[\begin{array}{l} t_0 := \tan x - x\\ t_1 := \frac{\sin x}{t_0} - \frac{x}{t_0}\\ \mathbf{if}\;x \leq -0.6406801188909744:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.06215681756197389:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (tan x) x)) (t_1 (- (/ (sin x) t_0) (/ x t_0))))
   (if (<= x -0.6406801188909744)
     t_1
     (if (<= x 0.06215681756197389)
       (+
        (+
         (* 0.225 (pow x 2.0))
         (+
          (* -0.009642857142857142 (pow x 4.0))
          (* 0.00024107142857142857 (pow x 6.0))))
        -0.5)
       t_1))))

C

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

↓

double code(double x) {
	double t_0 = tan(x) - x;
	double t_1 = (sin(x) / t_0) - (x / t_0);
	double tmp;
	if (x <= -0.6406801188909744) {
		tmp = t_1;
	} else if (x <= 0.06215681756197389) {
		tmp = ((0.225 * pow(x, 2.0)) + ((-0.009642857142857142 * pow(x, 4.0)) + (0.00024107142857142857 * pow(x, 6.0)))) + -0.5;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = tan(x) - x
    t_1 = (sin(x) / t_0) - (x / t_0)
    if (x <= (-0.6406801188909744d0)) then
        tmp = t_1
    else if (x <= 0.06215681756197389d0) then
        tmp = ((0.225d0 * (x ** 2.0d0)) + (((-0.009642857142857142d0) * (x ** 4.0d0)) + (0.00024107142857142857d0 * (x ** 6.0d0)))) + (-0.5d0)
    else
        tmp = t_1
    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.tan(x) - x;
	double t_1 = (Math.sin(x) / t_0) - (x / t_0);
	double tmp;
	if (x <= -0.6406801188909744) {
		tmp = t_1;
	} else if (x <= 0.06215681756197389) {
		tmp = ((0.225 * Math.pow(x, 2.0)) + ((-0.009642857142857142 * Math.pow(x, 4.0)) + (0.00024107142857142857 * Math.pow(x, 6.0)))) + -0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = math.tan(x) - x
	t_1 = (math.sin(x) / t_0) - (x / t_0)
	tmp = 0
	if x <= -0.6406801188909744:
		tmp = t_1
	elif x <= 0.06215681756197389:
		tmp = ((0.225 * math.pow(x, 2.0)) + ((-0.009642857142857142 * math.pow(x, 4.0)) + (0.00024107142857142857 * math.pow(x, 6.0)))) + -0.5
	else:
		tmp = t_1
	return tmp

Julia

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

↓

function code(x)
	t_0 = Float64(tan(x) - x)
	t_1 = Float64(Float64(sin(x) / t_0) - Float64(x / t_0))
	tmp = 0.0
	if (x <= -0.6406801188909744)
		tmp = t_1;
	elseif (x <= 0.06215681756197389)
		tmp = Float64(Float64(Float64(0.225 * (x ^ 2.0)) + Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(0.00024107142857142857 * (x ^ 6.0)))) + -0.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	t_0 = tan(x) - x;
	t_1 = (sin(x) / t_0) - (x / t_0);
	tmp = 0.0;
	if (x <= -0.6406801188909744)
		tmp = t_1;
	elseif (x <= 0.06215681756197389)
		tmp = ((0.225 * (x ^ 2.0)) + ((-0.009642857142857142 * (x ^ 4.0)) + (0.00024107142857142857 * (x ^ 6.0)))) + -0.5;
	else
		tmp = t_1;
	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[Tan[x], $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision] - N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.6406801188909744], t$95$1, If[LessEqual[x, 0.06215681756197389], N[(N[(N[(0.225 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.00024107142857142857 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.640680118890974426 or 0.062156817561973887 < 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

   3. Applied egg-rr 0.0

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

   if -0.640680118890974426 < x < 0.062156817561973887

   1. Initial program 63.0

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

   2. Simplified 63.0

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
      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. Applied egg-rr 63.5

      \[\leadsto \frac{\sin x - x}{\color{blue}{\log \left(e^{\tan x - x}\right)}} \]

   4. Taylor expanded in x around 0 0.0

      \[\leadsto \color{blue}{\left(0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.6406801188909744:\\ \;\;\;\;\frac{\sin x}{\tan x - x} - \frac{x}{\tan x - x}\\ \mathbf{elif}\;x \leq 0.06215681756197389:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{\tan x - x} - \frac{x}{\tan x - x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	20424

\[\begin{array}{l} t_0 := \tan x - x\\ t_1 := \frac{\sin x}{t_0} - \frac{x}{t_0}\\ \mathbf{if}\;x \leq -0.6406801188909744:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.06215681756197389:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.225, \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right) + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.1
Cost	20168

\[\begin{array}{l} t_0 := \tan x - x\\ t_1 := \frac{\sin x}{t_0} - \frac{x}{t_0}\\ \mathbf{if}\;x \leq -0.6406801188909744:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.192697744980965 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.009642857142857142, 0.225\right), -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.1
Cost	13640

\[\begin{array}{l} t_0 := \frac{\sin x - x}{\tan x - x}\\ \mathbf{if}\;x \leq -0.6406801188909744:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.192697744980965 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.009642857142857142, 0.225\right), -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.1
Cost	13640

\[\begin{array}{l} t_0 := \sin x - x\\ t_1 := \tan x - x\\ \mathbf{if}\;x \leq -0.6406801188909744:\\ \;\;\;\;\frac{t_0}{t_1}\\ \mathbf{elif}\;x \leq 9.192697744980965 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.009642857142857142, 0.225\right), -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_1}{t_0}}\\ \end{array} \]

Alternative 5
Error	0.1
Cost	13512

\[\begin{array}{l} t_0 := \frac{\sin x - x}{\tan x - x}\\ \mathbf{if}\;x \leq -0.008522875167759915:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.192697744980965 \cdot 10^{-7}:\\ \;\;\;\;-0.5 + x \cdot \left(x \cdot 0.225\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.7
Cost	7236

\[\begin{array}{l} t_0 := \tan x - x\\ \mathbf{if}\;x \leq -3.789803549298442:\\ \;\;\;\;\frac{3}{x \cdot x} - \frac{x}{t_0}\\ \mathbf{elif}\;x \leq 0.9677415055966984:\\ \;\;\;\;-0.5 + x \cdot \left(x \cdot 0.225\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{t_0}\\ \end{array} \]

Alternative 7
Error	0.7
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -3.789803549298442:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.9677415055966984:\\ \;\;\;\;-0.5 + x \cdot \left(x \cdot 0.225\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan x - x}\\ \end{array} \]

Alternative 8
Error	0.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -3.789803549298442:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.9677415055966984:\\ \;\;\;\;-0.5 + x \cdot \left(x \cdot 0.225\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	0.9
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -3.789803549298442:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.9677415055966984:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	31.6
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2022308 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))