sintan (problem 3.4.5)

Percentage Accurate: 50.4% → 99.5%

Time: 19.8s

Alternatives: 8

Speedup: 207.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x - \sin x}{x - \tan x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - \sin x}{x - \tan x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - \sin x}{x - \tan x} \leq 2:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin x - x}{\tan x - x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (- x (sin x)) (- x (tan x))) 2.0)
   (log1p (expm1 (/ (- (sin x) x) (- (tan x) x))))
   -0.5))

C

double code(double x) {
	double tmp;
	if (((x - sin(x)) / (x - tan(x))) <= 2.0) {
		tmp = log1p(expm1(((sin(x) - x) / (tan(x) - x))));
	} else {
		tmp = -0.5;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (((x - Math.sin(x)) / (x - Math.tan(x))) <= 2.0) {
		tmp = Math.log1p(Math.expm1(((Math.sin(x) - x) / (Math.tan(x) - x))));
	} else {
		tmp = -0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x - math.sin(x)) / (x - math.tan(x))) <= 2.0:
		tmp = math.log1p(math.expm1(((math.sin(x) - x) / (math.tan(x) - x))))
	else:
		tmp = -0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x - sin(x)) / Float64(x - tan(x))) <= 2.0)
		tmp = log1p(expm1(Float64(Float64(sin(x) - x) / Float64(tan(x) - x))));
	else
		tmp = -0.5;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[Log[1 + N[(Exp[N[(N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], -0.5]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (sin.f64 x)) (-.f64 x (tan.f64 x))) < 2

   1. Initial program 99.4%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. sub-neg 99.4%

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

      2. +-commutative 99.4%

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

      3. neg-sub0 99.4%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 99.4%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 99.4%

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

      6. neg-mul-1 99.4%

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

      7. sub-neg 99.4%

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

      8. +-commutative 99.4%

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

      9. neg-sub0 99.4%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 99.4%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 99.4%

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

      12. neg-mul-1 99.4%

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

      13. times-frac 99.4%

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

      14. metadata-eval 99.4%

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

      15. *-lft-identity 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
   4. Step-by-step derivation

      1. log1p-expm1-u 99.4%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin x - x}{\tan x - x}\right)\right)} \]

   5. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin x - x}{\tan x - x}\right)\right)} \]

   if 2 < (/.f64 (-.f64 x (sin.f64 x)) (-.f64 x (tan.f64 x)))

   1. Initial program 0.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. sub-neg 0.0%

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

      2. +-commutative 0.0%

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

      3. neg-sub0 0.0%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 0.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 0.0%

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

      6. neg-mul-1 0.0%

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

      7. sub-neg 0.0%

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

      8. +-commutative 0.0%

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

      9. neg-sub0 0.0%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 0.0%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 0.0%

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

      12. neg-mul-1 0.0%

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

      13. times-frac 0.0%

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

      14. metadata-eval 0.0%

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

      15. *-lft-identity 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \sin x}{x - \tan x} \leq 2:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\sin x - x}{\tan x - x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - \sin x}{x - \tan x} \leq 2:\\ \;\;\;\;\frac{1}{\frac{\tan x - x}{\sin x - x}}\\ \mathbf{else}:\\ \;\;\;\;-0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (- x (sin x)) (- x (tan x))) 2.0)
   (/ 1.0 (/ (- (tan x) x) (- (sin x) x)))
   -0.5))

C

double code(double x) {
	double tmp;
	if (((x - sin(x)) / (x - tan(x))) <= 2.0) {
		tmp = 1.0 / ((tan(x) - x) / (sin(x) - x));
	} else {
		tmp = -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((x - sin(x)) / (x - tan(x))) <= 2.0d0) then
        tmp = 1.0d0 / ((tan(x) - x) / (sin(x) - x))
    else
        tmp = -0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((x - Math.sin(x)) / (x - Math.tan(x))) <= 2.0) {
		tmp = 1.0 / ((Math.tan(x) - x) / (Math.sin(x) - x));
	} else {
		tmp = -0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((x - math.sin(x)) / (x - math.tan(x))) <= 2.0:
		tmp = 1.0 / ((math.tan(x) - x) / (math.sin(x) - x))
	else:
		tmp = -0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(x - sin(x)) / Float64(x - tan(x))) <= 2.0)
		tmp = Float64(1.0 / Float64(Float64(tan(x) - x) / Float64(sin(x) - x)));
	else
		tmp = -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((x - sin(x)) / (x - tan(x))) <= 2.0)
		tmp = 1.0 / ((tan(x) - x) / (sin(x) - x));
	else
		tmp = -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 / N[(N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (sin.f64 x)) (-.f64 x (tan.f64 x))) < 2

   1. Initial program 99.4%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. sub-neg 99.4%

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

      2. +-commutative 99.4%

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

      3. neg-sub0 99.4%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 99.4%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 99.4%

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

      6. neg-mul-1 99.4%

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

      7. sub-neg 99.4%

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

      8. +-commutative 99.4%

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

      9. neg-sub0 99.4%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 99.4%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 99.4%

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

      12. neg-mul-1 99.4%

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

      13. times-frac 99.4%

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

      14. metadata-eval 99.4%

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

      15. *-lft-identity 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
   4. Step-by-step derivation

      1. clear-num 99.4%

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

      2. inv-pow 99.4%

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

   5. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{{\left(\frac{\tan x - x}{\sin x - x}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 99.4%

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

   7. Applied egg-rr 99.4%

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

   if 2 < (/.f64 (-.f64 x (sin.f64 x)) (-.f64 x (tan.f64 x)))

   1. Initial program 0.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. sub-neg 0.0%

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

      2. +-commutative 0.0%

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

      3. neg-sub0 0.0%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 0.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 0.0%

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

      6. neg-mul-1 0.0%

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

      7. sub-neg 0.0%

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

      8. +-commutative 0.0%

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

      9. neg-sub0 0.0%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 0.0%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 0.0%

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

      12. neg-mul-1 0.0%

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

      13. times-frac 0.0%

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

      14. metadata-eval 0.0%

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

      15. *-lft-identity 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \sin x}{x - \tan x} \leq 2:\\ \;\;\;\;\frac{1}{\frac{\tan x - x}{\sin x - x}}\\ \mathbf{else}:\\ \;\;\;\;-0.5\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - \sin x}{x - \tan x}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- x (sin x)) (- x (tan x))))) (if (<= t_0 2.0) t_0 -0.5)))

C

double code(double x) {
	double t_0 = (x - sin(x)) / (x - tan(x));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - sin(x)) / (x - tan(x))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = -0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x - Math.sin(x)) / (x - Math.tan(x));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = -0.5;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x - math.sin(x)) / (x - math.tan(x))
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = -0.5
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x - sin(x)) / Float64(x - tan(x)))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x - sin(x)) / (x - tan(x));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, -0.5]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (sin.f64 x)) (-.f64 x (tan.f64 x))) < 2

   1. Initial program 99.4%

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

   if 2 < (/.f64 (-.f64 x (sin.f64 x)) (-.f64 x (tan.f64 x)))

   1. Initial program 0.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. sub-neg 0.0%

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

      2. +-commutative 0.0%

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

      3. neg-sub0 0.0%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 0.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 0.0%

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

      6. neg-mul-1 0.0%

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

      7. sub-neg 0.0%

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

      8. +-commutative 0.0%

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

      9. neg-sub0 0.0%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 0.0%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 0.0%

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

      12. neg-mul-1 0.0%

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

      13. times-frac 0.0%

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

      14. metadata-eval 0.0%

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

      15. *-lft-identity 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - \sin x}{x - \tan x} \leq 2:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;-0.5\\ \end{array} \]

Alternative 4: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;-0.5 + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{3}{x \cdot x} - \frac{x}{\tan x - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.6)
   1.0
   (if (<= x 2.8)
     (+ -0.5 (* 0.225 (* x x)))
     (- (/ 3.0 (* x x)) (/ x (- (tan x) x))))))

C

double code(double x) {
	double tmp;
	if (x <= -2.6) {
		tmp = 1.0;
	} else if (x <= 2.8) {
		tmp = -0.5 + (0.225 * (x * x));
	} else {
		tmp = (3.0 / (x * x)) - (x / (tan(x) - x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.6d0)) then
        tmp = 1.0d0
    else if (x <= 2.8d0) then
        tmp = (-0.5d0) + (0.225d0 * (x * x))
    else
        tmp = (3.0d0 / (x * x)) - (x / (tan(x) - x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.6) {
		tmp = 1.0;
	} else if (x <= 2.8) {
		tmp = -0.5 + (0.225 * (x * x));
	} else {
		tmp = (3.0 / (x * x)) - (x / (Math.tan(x) - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.6:
		tmp = 1.0
	elif x <= 2.8:
		tmp = -0.5 + (0.225 * (x * x))
	else:
		tmp = (3.0 / (x * x)) - (x / (math.tan(x) - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.6)
		tmp = 1.0;
	elseif (x <= 2.8)
		tmp = Float64(-0.5 + Float64(0.225 * Float64(x * x)));
	else
		tmp = Float64(Float64(3.0 / Float64(x * x)) - Float64(x / Float64(tan(x) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.6)
		tmp = 1.0;
	elseif (x <= 2.8)
		tmp = -0.5 + (0.225 * (x * x));
	else
		tmp = (3.0 / (x * x)) - (x / (tan(x) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.6], 1.0, If[LessEqual[x, 2.8], N[(-0.5 + N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.60000000000000009

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 100.0%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.8%

      \[\leadsto \color{blue}{1} \]

   if -2.60000000000000009 < x < 2.7999999999999998

   1. Initial program 3.3%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. sub-neg 3.3%

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

      2. +-commutative 3.3%

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

      3. neg-sub0 3.3%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 3.3%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 3.3%

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

      6. neg-mul-1 3.3%

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

      7. sub-neg 3.3%

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

      8. +-commutative 3.3%

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

      9. neg-sub0 3.3%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 3.3%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 3.3%

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

      12. neg-mul-1 3.3%

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

      13. times-frac 3.3%

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

      14. metadata-eval 3.3%

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

      15. *-lft-identity 3.3%

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

   3. Simplified 3.3%

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

   4. Taylor expanded in x around 0 98.9%

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - 0.5} \]
   5. Step-by-step derivation

      1. fma-neg 98.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, {x}^{2}, -0.5\right)} \]

      2. unpow2 98.9%

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

      3. metadata-eval 98.9%

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

   6. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)} \]
   7. Step-by-step derivation

      1. fma-udef 98.9%

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

   8. Applied egg-rr 98.9%

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

   if 2.7999999999999998 < x

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 100.0%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
   4. Step-by-step derivation

      1. div-sub 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 97.9%

      \[\leadsto \color{blue}{\frac{3}{{x}^{2}}} - \frac{x}{\tan x - x} \]
   7. Step-by-step derivation

      1. unpow2 97.9%

         \[\leadsto \frac{3}{\color{blue}{x \cdot x}} - \frac{x}{\tan x - x} \]

   8. Simplified 97.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;-0.5 + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{3}{x \cdot x} - \frac{x}{\tan x - x}\\ \end{array} \]

Alternative 5: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;-0.5 + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan x - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.6)
   1.0
   (if (<= x 1.45) (+ -0.5 (* 0.225 (* x x))) (/ (- x) (- (tan x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -2.6) {
		tmp = 1.0;
	} else if (x <= 1.45) {
		tmp = -0.5 + (0.225 * (x * x));
	} else {
		tmp = -x / (tan(x) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.6d0)) then
        tmp = 1.0d0
    else if (x <= 1.45d0) then
        tmp = (-0.5d0) + (0.225d0 * (x * x))
    else
        tmp = -x / (tan(x) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.6) {
		tmp = 1.0;
	} else if (x <= 1.45) {
		tmp = -0.5 + (0.225 * (x * x));
	} else {
		tmp = -x / (Math.tan(x) - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.6:
		tmp = 1.0
	elif x <= 1.45:
		tmp = -0.5 + (0.225 * (x * x))
	else:
		tmp = -x / (math.tan(x) - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.6)
		tmp = 1.0;
	elseif (x <= 1.45)
		tmp = Float64(-0.5 + Float64(0.225 * Float64(x * x)));
	else
		tmp = Float64(Float64(-x) / Float64(tan(x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.6)
		tmp = 1.0;
	elseif (x <= 1.45)
		tmp = -0.5 + (0.225 * (x * x));
	else
		tmp = -x / (tan(x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.6], 1.0, If[LessEqual[x, 1.45], N[(-0.5 + N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-x) / N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.60000000000000009

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 100.0%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.8%

      \[\leadsto \color{blue}{1} \]

   if -2.60000000000000009 < x < 1.44999999999999996

   1. Initial program 3.3%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. sub-neg 3.3%

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

      2. +-commutative 3.3%

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

      3. neg-sub0 3.3%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 3.3%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 3.3%

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

      6. neg-mul-1 3.3%

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

      7. sub-neg 3.3%

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

      8. +-commutative 3.3%

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

      9. neg-sub0 3.3%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 3.3%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 3.3%

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

      12. neg-mul-1 3.3%

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

      13. times-frac 3.3%

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

      14. metadata-eval 3.3%

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

      15. *-lft-identity 3.3%

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

   3. Simplified 3.3%

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

   4. Taylor expanded in x around 0 98.9%

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - 0.5} \]
   5. Step-by-step derivation

      1. fma-neg 98.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, {x}^{2}, -0.5\right)} \]

      2. unpow2 98.9%

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

      3. metadata-eval 98.9%

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

   6. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)} \]
   7. Step-by-step derivation

      1. fma-udef 98.9%

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

   8. Applied egg-rr 98.9%

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

   if 1.44999999999999996 < x

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 100.0%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
   4. Step-by-step derivation

      1. flip-- 47.3%

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

      2. div-inv 47.1%

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

      3. pow2 47.1%

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

      4. +-commutative 47.1%

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

   5. Applied egg-rr 47.1%

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

      1. associate-*r/ 47.3%

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

      2. *-rgt-identity 47.3%

         \[\leadsto \frac{\frac{\color{blue}{{\sin x}^{2} - x \cdot x}}{x + \sin x}}{\tan x - x} \]

   7. Simplified 47.3%

      \[\leadsto \frac{\color{blue}{\frac{{\sin x}^{2} - x \cdot x}{x + \sin x}}}{\tan x - x} \]

   8. Taylor expanded in x around inf 97.9%

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

      1. neg-mul-1 97.9%

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

   10. Simplified 97.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;-0.5 + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan x - x}\\ \end{array} \]

Alternative 6: 98.8% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;-0.5 + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.6) 1.0 (if (<= x 2.5) (+ -0.5 (* 0.225 (* x x))) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -2.6) {
		tmp = 1.0;
	} else if (x <= 2.5) {
		tmp = -0.5 + (0.225 * (x * x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.6d0)) then
        tmp = 1.0d0
    else if (x <= 2.5d0) then
        tmp = (-0.5d0) + (0.225d0 * (x * x))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.6) {
		tmp = 1.0;
	} else if (x <= 2.5) {
		tmp = -0.5 + (0.225 * (x * x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.6:
		tmp = 1.0
	elif x <= 2.5:
		tmp = -0.5 + (0.225 * (x * x))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.6)
		tmp = 1.0;
	elseif (x <= 2.5)
		tmp = Float64(-0.5 + Float64(0.225 * Float64(x * x)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.6)
		tmp = 1.0;
	elseif (x <= 2.5)
		tmp = -0.5 + (0.225 * (x * x));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.6], 1.0, If[LessEqual[x, 2.5], N[(-0.5 + N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000009 or 2.5 < x

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 100.0%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.3%

      \[\leadsto \color{blue}{1} \]

   if -2.60000000000000009 < x < 2.5

   1. Initial program 3.3%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. sub-neg 3.3%

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

      2. +-commutative 3.3%

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

      3. neg-sub0 3.3%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 3.3%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 3.3%

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

      6. neg-mul-1 3.3%

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

      7. sub-neg 3.3%

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

      8. +-commutative 3.3%

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

      9. neg-sub0 3.3%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 3.3%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 3.3%

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

      12. neg-mul-1 3.3%

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

      13. times-frac 3.3%

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

      14. metadata-eval 3.3%

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

      15. *-lft-identity 3.3%

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

   3. Simplified 3.3%

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

   4. Taylor expanded in x around 0 98.9%

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - 0.5} \]
   5. Step-by-step derivation

      1. fma-neg 98.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, {x}^{2}, -0.5\right)} \]

      2. unpow2 98.9%

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

      3. metadata-eval 98.9%

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

   6. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)} \]
   7. Step-by-step derivation

      1. fma-udef 98.9%

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

   8. Applied egg-rr 98.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;-0.5 + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 98.5% accurate, 40.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -1.6) 1.0 (if (<= x 1.6) -0.5 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.6) {
		tmp = 1.0;
	} else if (x <= 1.6) {
		tmp = -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.6d0)) then
        tmp = 1.0d0
    else if (x <= 1.6d0) then
        tmp = -0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.6) {
		tmp = 1.0;
	} else if (x <= 1.6) {
		tmp = -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.6:
		tmp = 1.0
	elif x <= 1.6:
		tmp = -0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.6)
		tmp = 1.0;
	elseif (x <= 1.6)
		tmp = -0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.6)
		tmp = 1.0;
	elseif (x <= 1.6)
		tmp = -0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.6], 1.0, If[LessEqual[x, 1.6], -0.5, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001 or 1.6000000000000001 < x

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 100.0%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.3%

      \[\leadsto \color{blue}{1} \]

   if -1.6000000000000001 < x < 1.6000000000000001

   1. Initial program 3.3%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. sub-neg 3.3%

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

      2. +-commutative 3.3%

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

      3. neg-sub0 3.3%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 3.3%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 3.3%

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

      6. neg-mul-1 3.3%

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

      7. sub-neg 3.3%

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

      8. +-commutative 3.3%

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

      9. neg-sub0 3.3%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 3.3%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 3.3%

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

      12. neg-mul-1 3.3%

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

      13. times-frac 3.3%

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

      14. metadata-eval 3.3%

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

      15. *-lft-identity 3.3%

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

   3. Simplified 3.3%

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

   4. Taylor expanded in x around 0 98.0%

      \[\leadsto \color{blue}{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 50.7% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -0.5)

C

double code(double x) {
	return -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -0.5d0
end function

Java

public static double code(double x) {
	return -0.5;
}

Python

def code(x):
	return -0.5

Julia

function code(x)
	return -0.5
end

MATLAB

function tmp = code(x)
	tmp = -0.5;
end

Wolfram

code[x_] := -0.5

Derivation

1. Initial program 52.0%

   \[\frac{x - \sin x}{x - \tan x} \]
2. Step-by-step derivation

   1. sub-neg 52.0%

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

   2. +-commutative 52.0%

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

   3. neg-sub0 52.0%

      \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

   4. associate-+l- 52.0%

      \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

   5. sub0-neg 52.0%

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

   6. neg-mul-1 52.0%

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

   7. sub-neg 52.0%

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

   8. +-commutative 52.0%

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

   9. neg-sub0 52.0%

      \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

   10. associate-+l- 52.0%

       \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

   11. sub0-neg 52.0%

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

   12. neg-mul-1 52.0%

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

   13. times-frac 52.0%

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

   14. metadata-eval 52.0%

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

   15. *-lft-identity 52.0%

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

3. Simplified 52.0%

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

4. Taylor expanded in x around 0 49.4%

   \[\leadsto \color{blue}{-0.5} \]

5. Final simplification 49.4%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023182 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))