sintan (problem 3.4.5)

Average Error: 32.3 → 0.3

Time: 21.5s

Precision: binary64

Cost: 13768

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4:\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \mathbf{elif}\;x \leq 0.0052:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\tan x - x}}{\frac{1}{\sin x - x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -2.4)
   (+ 1.0 (/ (- (tan x) (sin x)) x))
   (if (<= x 0.0052)
     (+ (* x (* x 0.225)) -0.5)
     (/ (/ 1.0 (- (tan x) x)) (/ 1.0 (- (sin x) x))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -2.4) {
		tmp = 1.0 + ((tan(x) - sin(x)) / x);
	} else if (x <= 0.0052) {
		tmp = (x * (x * 0.225)) + -0.5;
	} else {
		tmp = (1.0 / (tan(x) - x)) / (1.0 / (sin(x) - x));
	}
	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) :: tmp
    if (x <= (-2.4d0)) then
        tmp = 1.0d0 + ((tan(x) - sin(x)) / x)
    else if (x <= 0.0052d0) then
        tmp = (x * (x * 0.225d0)) + (-0.5d0)
    else
        tmp = (1.0d0 / (tan(x) - x)) / (1.0d0 / (sin(x) - x))
    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 tmp;
	if (x <= -2.4) {
		tmp = 1.0 + ((Math.tan(x) - Math.sin(x)) / x);
	} else if (x <= 0.0052) {
		tmp = (x * (x * 0.225)) + -0.5;
	} else {
		tmp = (1.0 / (Math.tan(x) - x)) / (1.0 / (Math.sin(x) - x));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -2.4:
		tmp = 1.0 + ((math.tan(x) - math.sin(x)) / x)
	elif x <= 0.0052:
		tmp = (x * (x * 0.225)) + -0.5
	else:
		tmp = (1.0 / (math.tan(x) - x)) / (1.0 / (math.sin(x) - x))
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -2.4)
		tmp = Float64(1.0 + Float64(Float64(tan(x) - sin(x)) / x));
	elseif (x <= 0.0052)
		tmp = Float64(Float64(x * Float64(x * 0.225)) + -0.5);
	else
		tmp = Float64(Float64(1.0 / Float64(tan(x) - x)) / Float64(1.0 / Float64(sin(x) - x)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.4)
		tmp = 1.0 + ((tan(x) - sin(x)) / x);
	elseif (x <= 0.0052)
		tmp = (x * (x * 0.225)) + -0.5;
	else
		tmp = (1.0 / (tan(x) - x)) / (1.0 / (sin(x) - x));
	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_] := If[LessEqual[x, -2.4], N[(1.0 + N[(N[(N[Tan[x], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0052], N[(N[(x * N[(x * 0.225), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(N[(1.0 / N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.39999999999999991

   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

      [Start] 0.0	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 0.0	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 0.0	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 0.0	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 0.0	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 0.0	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 0.0	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 0.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 0.0	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 0.0	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 0.0	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Taylor expanded in x around -inf 0.6

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{\sin x - \frac{\sin x}{\cos x}}{x}} \]

   4. Simplified 0.6

      \[\leadsto \color{blue}{1 + \frac{\frac{\sin x}{\cos x} + \left(-\sin x\right)}{x}} \]
      Proof

      [Start] 0.6	\[ 1 + -1 \cdot \frac{\sin x - \frac{\sin x}{\cos x}}{x} \]
      associate-*r/ [=>] 0.6	\[ 1 + \color{blue}{\frac{-1 \cdot \left(\sin x - \frac{\sin x}{\cos x}\right)}{x}} \]
      distribute-lft-out-- [<=] 0.6	\[ 1 + \frac{\color{blue}{-1 \cdot \sin x - -1 \cdot \frac{\sin x}{\cos x}}}{x} \]
      cancel-sign-sub-inv [=>] 0.6	\[ 1 + \frac{\color{blue}{-1 \cdot \sin x + \left(--1\right) \cdot \frac{\sin x}{\cos x}}}{x} \]
      +-commutative [=>] 0.6	\[ 1 + \frac{\color{blue}{\left(--1\right) \cdot \frac{\sin x}{\cos x} + -1 \cdot \sin x}}{x} \]
      metadata-eval [=>] 0.6	\[ 1 + \frac{\color{blue}{1} \cdot \frac{\sin x}{\cos x} + -1 \cdot \sin x}{x} \]
      *-commutative [=>] 0.6	\[ 1 + \frac{\color{blue}{\frac{\sin x}{\cos x} \cdot 1} + -1 \cdot \sin x}{x} \]
      associate-/r/ [<=] 0.6	\[ 1 + \frac{\color{blue}{\frac{\sin x}{\frac{\cos x}{1}}} + -1 \cdot \sin x}{x} \]
      /-rgt-identity [=>] 0.6	\[ 1 + \frac{\frac{\sin x}{\color{blue}{\cos x}} + -1 \cdot \sin x}{x} \]
      mul-1-neg [=>] 0.6	\[ 1 + \frac{\frac{\sin x}{\cos x} + \color{blue}{\left(-\sin x\right)}}{x} \]

   5. Applied egg-rr 0.6

      \[\leadsto 1 + \frac{\color{blue}{\tan x} + \left(-\sin x\right)}{x} \]

   6. Applied egg-rr 16.1

      \[\leadsto 1 + \frac{\color{blue}{e^{\mathsf{log1p}\left(\tan x + \left(-\sin x\right)\right)} - 1}}{x} \]

   7. Simplified 0.6

      \[\leadsto 1 + \frac{\color{blue}{\tan x - \sin x}}{x} \]
      Proof

      [Start] 16.1	\[ 1 + \frac{e^{\mathsf{log1p}\left(\tan x + \left(-\sin x\right)\right)} - 1}{x} \]
      expm1-def [=>] 16.1	\[ 1 + \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan x + \left(-\sin x\right)\right)\right)}}{x} \]
      expm1-log1p [=>] 0.6	\[ 1 + \frac{\color{blue}{\tan x + \left(-\sin x\right)}}{x} \]
      unsub-neg [=>] 0.6	\[ 1 + \frac{\color{blue}{\tan x - \sin x}}{x} \]

   if -2.39999999999999991 < x < 0.0051999999999999998

   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

      [Start] 63.0	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 63.0	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 63.0	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 63.0	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 63.0	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 63.0	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 63.0	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 63.0	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 63.0	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 63.0	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 63.0	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Taylor expanded in x around 0 0.2

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - 0.5} \]

   4. Simplified 0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)} \]
      Proof

      [Start] 0.2	\[ 0.225 \cdot {x}^{2} - 0.5 \]
      unpow2 [=>] 0.2	\[ 0.225 \cdot \color{blue}{\left(x \cdot x\right)} - 0.5 \]
      fma-neg [=>] 0.2	\[ \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)} \]
      metadata-eval [=>] 0.2	\[ \mathsf{fma}\left(0.225, x \cdot x, \color{blue}{-0.5}\right) \]

   5. Applied egg-rr 0.2

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

   if 0.0051999999999999998 < x

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \frac{x - \sin x}{x - \tan x} \]
      sub-neg [=>] 0.1	\[ \frac{\color{blue}{x + \left(-\sin x\right)}}{x - \tan x} \]
      +-commutative [=>] 0.1	\[ \frac{\color{blue}{\left(-\sin x\right) + x}}{x - \tan x} \]
      neg-sub0 [=>] 0.1	\[ \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]
      associate-+l- [=>] 0.1	\[ \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]
      sub0-neg [=>] 0.1	\[ \frac{\color{blue}{-\left(\sin x - x\right)}}{x - \tan x} \]
      neg-mul-1 [=>] 0.1	\[ \frac{\color{blue}{-1 \cdot \left(\sin x - x\right)}}{x - \tan x} \]
      sub-neg [=>] 0.1	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{x + \left(-\tan x\right)}} \]
      +-commutative [=>] 0.1	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(-\tan x\right) + x}} \]
      neg-sub0 [=>] 0.1	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]
      associate-+l- [=>] 0.1	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]
      sub0-neg [=>] 0.1	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-\left(\tan x - x\right)}} \]
      neg-mul-1 [=>] 0.1	\[ \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{-1 \cdot \left(\tan x - x\right)}} \]
      times-frac [=>] 0.1	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{\sin x - x}{\tan x - x}} \]
      metadata-eval [=>] 0.1	\[ \color{blue}{1} \cdot \frac{\sin x - x}{\tan x - x} \]
      *-lft-identity [=>] 0.1	\[ \color{blue}{\frac{\sin x - x}{\tan x - x}} \]

   3. Applied egg-rr 0.2

      \[\leadsto \color{blue}{\frac{1}{\tan x - x} \cdot \left(\sin x - x\right)} \]

   4. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\frac{\frac{1}{\tan x - x}}{\frac{1}{\sin x - x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4:\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \mathbf{elif}\;x \leq 0.0052:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\tan x - x}}{\frac{1}{\sin x - x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	13513

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \lor \neg \left(x \leq 2.4\right):\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \end{array} \]

Alternative 2
Error	0.3
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4:\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \mathbf{elif}\;x \leq 0.0052:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 3
Error	0.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	0.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9:\\ \;\;\;\;1 + \frac{3}{x \cdot x}\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	1.0
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.56:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	31.0
Cost	64

\[-0.5 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))