sintan (problem 3.4.5)

Percentage Accurate: 51.4% → 99.6%

Time: 14.7s

Alternatives: 7

Speedup: 67.6×

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: 51.4% 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: 99.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = tan(x) - x;
	double tmp;
	if (((x - sin(x)) / (x - tan(x))) <= 2.0) {
		tmp = (sin(x) / t_0) - (x / 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 = tan(x) - x
    if (((x - sin(x)) / (x - tan(x))) <= 2.0d0) then
        tmp = (sin(x) / t_0) - (x / t_0)
    else
        tmp = -0.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   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 100.0%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((x - sin(x)) / (x - tan(x))) <= 2.0) {
		tmp = pow(((tan(x) - x) / (sin(x) - x)), -1.0);
	} 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 = ((tan(x) - x) / (sin(x) - x)) ** (-1.0d0)
    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 = Math.pow(((Math.tan(x) - x) / (Math.sin(x) - x)), -1.0);
	} 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.pow(((math.tan(x) - x) / (math.sin(x) - x)), -1.0)
	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(Float64(tan(x) - x) / Float64(sin(x) - x)) ^ -1.0;
	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 = ((tan(x) - x) / (sin(x) - x)) ^ -1.0;
	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[Power[N[(N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], -1.0], $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 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. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   5. Applied egg-rr 100.0%

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

   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 100.0%

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

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 100.0%

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

   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 100.0%

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

Alternative 4: 74.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{3}{x \cdot x} - \frac{x}{\tan x - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55) -0.5 (- (/ 3.0 (* x x)) (/ x (- (tan x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = -0.5;
	} 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 <= 1.55d0) then
        tmp = -0.5d0
    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 <= 1.55) {
		tmp = -0.5;
	} else {
		tmp = (3.0 / (x * x)) - (x / (Math.tan(x) - x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = -0.5
	else:
		tmp = (3.0 / (x * x)) - (x / (math.tan(x) - x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = -0.5;
	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 <= 1.55)
		tmp = -0.5;
	else
		tmp = (3.0 / (x * x)) - (x / (tan(x) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], -0.5, 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 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 35.8%

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

      1. sub-neg 35.8%

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

      2. +-commutative 35.8%

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

      3. neg-sub0 35.8%

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

      4. associate-+l- 35.8%

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

      5. sub0-neg 35.8%

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

      6. neg-mul-1 35.8%

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

      7. sub-neg 35.8%

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

      8. +-commutative 35.8%

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

      9. neg-sub0 35.8%

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

      10. associate-+l- 35.8%

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

      11. sub0-neg 35.8%

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

      12. neg-mul-1 35.8%

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

      13. times-frac 35.8%

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

      14. metadata-eval 35.8%

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

      15. *-lft-identity 35.8%

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

   3. Simplified 35.8%

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

   4. Taylor expanded in x around 0 64.7%

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

   if 1.55000000000000004 < x

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. neg-sub0 99.9%

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

      4. associate-+l- 99.9%

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

      5. sub0-neg 99.9%

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

      6. neg-mul-1 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. neg-sub0 99.9%

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

      10. associate-+l- 99.9%

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

      11. sub0-neg 99.9%

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

      12. neg-mul-1 99.9%

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

      13. times-frac 99.9%

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

      14. metadata-eval 99.9%

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

      15. *-lft-identity 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\sin x - x}{\tan x - x}} \]
   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 99.2%

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

      1. unpow2 99.2%

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

   8. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{3}{x \cdot x}} - \frac{x}{\tan x - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{3}{x \cdot x} - \frac{x}{\tan x - x}\\ \end{array} \]

Alternative 5: 74.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.55) -0.5 (/ (- x (sin x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = -0.5;
	} else {
		tmp = (x - sin(x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.55d0) then
        tmp = -0.5d0
    else
        tmp = (x - sin(x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = -0.5;
	} else {
		tmp = (x - Math.sin(x)) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = -0.5
	else:
		tmp = (x - math.sin(x)) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = -0.5;
	else
		tmp = Float64(Float64(x - sin(x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.55)
		tmp = -0.5;
	else
		tmp = (x - sin(x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], -0.5, N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 35.8%

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

      1. sub-neg 35.8%

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

      2. +-commutative 35.8%

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

      3. neg-sub0 35.8%

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

      4. associate-+l- 35.8%

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

      5. sub0-neg 35.8%

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

      6. neg-mul-1 35.8%

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

      7. sub-neg 35.8%

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

      8. +-commutative 35.8%

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

      9. neg-sub0 35.8%

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

      10. associate-+l- 35.8%

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

      11. sub0-neg 35.8%

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

      12. neg-mul-1 35.8%

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

      13. times-frac 35.8%

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

      14. metadata-eval 35.8%

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

      15. *-lft-identity 35.8%

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

   3. Simplified 35.8%

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

   4. Taylor expanded in x around 0 64.7%

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

   if 1.55000000000000004 < x

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. neg-sub0 99.9%

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

      4. associate-+l- 99.9%

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

      5. sub0-neg 99.9%

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

      6. neg-mul-1 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. neg-sub0 99.9%

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

      10. associate-+l- 99.9%

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

      11. sub0-neg 99.9%

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

      12. neg-mul-1 99.9%

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

      13. times-frac 99.9%

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

      14. metadata-eval 99.9%

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

      15. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 99.1%

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

      1. neg-mul-1 99.1%

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

   6. Simplified 99.1%

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

      1. frac-2neg 99.1%

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

      2. div-inv 98.9%

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

      3. sub-neg 98.9%

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

      4. distribute-neg-in 98.9%

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

      5. remove-double-neg 98.9%

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

      6. remove-double-neg 98.9%

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

   8. Applied egg-rr 98.9%

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

      1. associate-*r/ 99.1%

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

      2. *-rgt-identity 99.1%

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

      3. +-commutative 99.1%

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

      4. unsub-neg 99.1%

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

   10. Simplified 99.1%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x}\\ \end{array} \]

Alternative 6: 74.2% accurate, 67.6× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.55) -0.5 1.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		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 <= 1.55d0) 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 <= 1.55) {
		tmp = -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = -0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = -0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.55)
		tmp = -0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], -0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 35.8%

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

      1. sub-neg 35.8%

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

      2. +-commutative 35.8%

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

      3. neg-sub0 35.8%

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

      4. associate-+l- 35.8%

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

      5. sub0-neg 35.8%

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

      6. neg-mul-1 35.8%

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

      7. sub-neg 35.8%

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

      8. +-commutative 35.8%

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

      9. neg-sub0 35.8%

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

      10. associate-+l- 35.8%

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

      11. sub0-neg 35.8%

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

      12. neg-mul-1 35.8%

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

      13. times-frac 35.8%

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

      14. metadata-eval 35.8%

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

      15. *-lft-identity 35.8%

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

   3. Simplified 35.8%

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

   4. Taylor expanded in x around 0 64.7%

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

   if 1.55000000000000004 < x

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. neg-sub0 99.9%

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

      4. associate-+l- 99.9%

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

      5. sub0-neg 99.9%

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

      6. neg-mul-1 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. neg-sub0 99.9%

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

      10. associate-+l- 99.9%

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

      11. sub0-neg 99.9%

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

      12. neg-mul-1 99.9%

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

      13. times-frac 99.9%

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

      14. metadata-eval 99.9%

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

      15. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 99.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

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

Alternative 7: 49.8% 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 53.1%

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

   1. sub-neg 53.1%

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

   2. +-commutative 53.1%

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

   3. neg-sub0 53.1%

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

   4. associate-+l- 53.1%

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

   5. sub0-neg 53.1%

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

   6. neg-mul-1 53.1%

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

   7. sub-neg 53.1%

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

   8. +-commutative 53.1%

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

   9. neg-sub0 53.1%

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

   10. associate-+l- 53.1%

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

   11. sub0-neg 53.1%

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

   12. neg-mul-1 53.1%

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

   13. times-frac 53.1%

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

   14. metadata-eval 53.1%

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

   15. *-lft-identity 53.1%

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

3. Simplified 53.1%

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

4. Taylor expanded in x around 0 47.7%

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

5. Final simplification 47.7%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023261 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))