sintan (problem 3.4.5)

Percentage Accurate: 50.7% → 99.9%

Time: 17.2s

Alternatives: 8

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: 50.7% 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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \tan x - x\\ \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{t_0} - \frac{x}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (tan x) x)))
   (if (<= x 0.0058)
     (+ (* 0.225 (* x x)) -0.5)
     (- (/ (sin x) t_0) (/ x t_0)))))

C

x = abs(x);
double code(double x) {
	double t_0 = tan(x) - x;
	double tmp;
	if (x <= 0.0058) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = (sin(x) / t_0) - (x / t_0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(x) - x
    if (x <= 0.0058d0) then
        tmp = (0.225d0 * (x * x)) + (-0.5d0)
    else
        tmp = (sin(x) / t_0) - (x / t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = Math.tan(x) - x;
	double tmp;
	if (x <= 0.0058) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = (Math.sin(x) / t_0) - (x / t_0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = math.tan(x) - x
	tmp = 0
	if x <= 0.0058:
		tmp = (0.225 * (x * x)) + -0.5
	else:
		tmp = (math.sin(x) / t_0) - (x / t_0)
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(tan(x) - x)
	tmp = 0.0
	if (x <= 0.0058)
		tmp = Float64(Float64(0.225 * Float64(x * x)) + -0.5);
	else
		tmp = Float64(Float64(sin(x) / t_0) - Float64(x / t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = tan(x) - x;
	tmp = 0.0;
	if (x <= 0.0058)
		tmp = (0.225 * (x * x)) + -0.5;
	else
		tmp = (sin(x) / t_0) - (x / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, 0.0058], N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision] - N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0058

   1. Initial program 34.1%

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

      1. sub-neg 34.1%

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

      2. +-commutative 34.1%

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

      3. neg-sub0 34.1%

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

      4. associate-+l- 34.1%

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

      5. sub0-neg 34.1%

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

      6. neg-mul-1 34.1%

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

      7. sub-neg 34.1%

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

      8. +-commutative 34.1%

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

      9. neg-sub0 34.1%

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

      10. associate-+l- 34.1%

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

      11. sub0-neg 34.1%

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

      12. neg-mul-1 34.1%

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

      13. times-frac 34.1%

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

      14. metadata-eval 34.1%

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

      15. *-lft-identity 34.1%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in x around 0 68.4%

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

      1. fma-neg 68.4%

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

      2. unpow2 68.4%

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

      3. metadata-eval 68.4%

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

   6. Simplified 68.4%

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

      1. fma-udef 68.4%

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

   8. Applied egg-rr 68.4%

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

   if 0.0058 < 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 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0058:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{\tan x - x} - \frac{x}{\tan x - x}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\tan x - x}{\sin x - x}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0044)
   (+ (* 0.225 (* x x)) -0.5)
   (pow (/ (- (tan x) x) (- (sin x) x)) -1.0)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0044) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = pow(((tan(x) - x) / (sin(x) - x)), -1.0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0044d0) then
        tmp = (0.225d0 * (x * x)) + (-0.5d0)
    else
        tmp = ((tan(x) - x) / (sin(x) - x)) ** (-1.0d0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.0044) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = Math.pow(((Math.tan(x) - x) / (Math.sin(x) - x)), -1.0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.0044:
		tmp = (0.225 * (x * x)) + -0.5
	else:
		tmp = math.pow(((math.tan(x) - x) / (math.sin(x) - x)), -1.0)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.0044)
		tmp = Float64(Float64(0.225 * Float64(x * x)) + -0.5);
	else
		tmp = Float64(Float64(tan(x) - x) / Float64(sin(x) - x)) ^ -1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0044)
		tmp = (0.225 * (x * x)) + -0.5;
	else
		tmp = ((tan(x) - x) / (sin(x) - x)) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.0044], N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[Power[N[(N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00440000000000000027

   1. Initial program 34.1%

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

      1. sub-neg 34.1%

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

      2. +-commutative 34.1%

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

      3. neg-sub0 34.1%

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

      4. associate-+l- 34.1%

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

      5. sub0-neg 34.1%

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

      6. neg-mul-1 34.1%

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

      7. sub-neg 34.1%

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

      8. +-commutative 34.1%

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

      9. neg-sub0 34.1%

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

      10. associate-+l- 34.1%

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

      11. sub0-neg 34.1%

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

      12. neg-mul-1 34.1%

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

      13. times-frac 34.1%

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

      14. metadata-eval 34.1%

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

      15. *-lft-identity 34.1%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in x around 0 68.4%

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

      1. fma-neg 68.4%

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

      2. unpow2 68.4%

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

      3. metadata-eval 68.4%

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

   6. Simplified 68.4%

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

      1. fma-udef 68.4%

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

   8. Applied egg-rr 68.4%

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

   if 0.00440000000000000027 < 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. clear-num 99.9%

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

      2. inv-pow 99.9%

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

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left(\frac{\tan x - x}{\sin x - x}\right)}^{-1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\tan x - x}{\sin x - x}\right)}^{-1}\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0044) (+ (* 0.225 (* x x)) -0.5) (/ (- x (sin x)) (- x (tan x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0044) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = (x - sin(x)) / (x - tan(x));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0044d0) then
        tmp = (0.225d0 * (x * x)) + (-0.5d0)
    else
        tmp = (x - sin(x)) / (x - tan(x))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.0044) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = (x - Math.sin(x)) / (x - Math.tan(x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.0044:
		tmp = (0.225 * (x * x)) + -0.5
	else:
		tmp = (x - math.sin(x)) / (x - math.tan(x))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.0044)
		tmp = Float64(Float64(0.225 * Float64(x * x)) + -0.5);
	else
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0044)
		tmp = (0.225 * (x * x)) + -0.5;
	else
		tmp = (x - sin(x)) / (x - tan(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.0044], N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00440000000000000027

   1. Initial program 34.1%

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

      1. sub-neg 34.1%

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

      2. +-commutative 34.1%

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

      3. neg-sub0 34.1%

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

      4. associate-+l- 34.1%

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

      5. sub0-neg 34.1%

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

      6. neg-mul-1 34.1%

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

      7. sub-neg 34.1%

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

      8. +-commutative 34.1%

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

      9. neg-sub0 34.1%

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

      10. associate-+l- 34.1%

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

      11. sub0-neg 34.1%

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

      12. neg-mul-1 34.1%

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

      13. times-frac 34.1%

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

      14. metadata-eval 34.1%

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

      15. *-lft-identity 34.1%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in x around 0 68.4%

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

      1. fma-neg 68.4%

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

      2. unpow2 68.4%

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

      3. metadata-eval 68.4%

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

   6. Simplified 68.4%

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

      1. fma-udef 68.4%

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

   8. Applied egg-rr 68.4%

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

   if 0.00440000000000000027 < x

   1. Initial program 99.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 4: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{x + \left(\tan x - \left(x + x\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (+ (* 0.225 (* x x)) -0.5)
   (/ (- x) (+ x (- (tan x) (+ x x))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = -x / (x + (tan(x) - (x + x)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.2d0) then
        tmp = (0.225d0 * (x * x)) + (-0.5d0)
    else
        tmp = -x / (x + (tan(x) - (x + x)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = -x / (x + (Math.tan(x) - (x + x)));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = (0.225 * (x * x)) + -0.5
	else:
		tmp = -x / (x + (math.tan(x) - (x + x)))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64(Float64(0.225 * Float64(x * x)) + -0.5);
	else
		tmp = Float64(Float64(-x) / Float64(x + Float64(tan(x) - Float64(x + x))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = (0.225 * (x * x)) + -0.5;
	else
		tmp = -x / (x + (tan(x) - (x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.2], N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[((-x) / N[(x + N[(N[Tan[x], $MachinePrecision] - N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 34.1%

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

      1. sub-neg 34.1%

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

      2. +-commutative 34.1%

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

      3. neg-sub0 34.1%

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

      4. associate-+l- 34.1%

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

      5. sub0-neg 34.1%

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

      6. neg-mul-1 34.1%

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

      7. sub-neg 34.1%

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

      8. +-commutative 34.1%

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

      9. neg-sub0 34.1%

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

      10. associate-+l- 34.1%

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

      11. sub0-neg 34.1%

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

      12. neg-mul-1 34.1%

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

      13. times-frac 34.1%

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

      14. metadata-eval 34.1%

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

      15. *-lft-identity 34.1%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in x around 0 68.4%

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

      1. fma-neg 68.4%

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

      2. unpow2 68.4%

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

      3. metadata-eval 68.4%

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

   6. Simplified 68.4%

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

      1. fma-udef 68.4%

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

   8. Applied egg-rr 68.4%

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

   if 2.2000000000000002 < 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 95.0%

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

      1. neg-mul-1 95.0%

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

   6. Simplified 95.0%

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

      1. add-sqr-sqrt 50.9%

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

      2. *-un-lft-identity 50.9%

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

      3. prod-diff 50.9%

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{fma}\left(\sqrt{\tan x}, \sqrt{\tan x}, -x \cdot 1\right) + \mathsf{fma}\left(-x, 1, x \cdot 1\right)}} \]

      4. *-commutative 50.9%

         \[\leadsto \frac{-x}{\mathsf{fma}\left(\sqrt{\tan x}, \sqrt{\tan x}, -\color{blue}{1 \cdot x}\right) + \mathsf{fma}\left(-x, 1, x \cdot 1\right)} \]

      5. *-un-lft-identity 50.9%

         \[\leadsto \frac{-x}{\mathsf{fma}\left(\sqrt{\tan x}, \sqrt{\tan x}, -\color{blue}{x}\right) + \mathsf{fma}\left(-x, 1, x \cdot 1\right)} \]

      6. fma-neg 50.9%

         \[\leadsto \frac{-x}{\color{blue}{\left(\sqrt{\tan x} \cdot \sqrt{\tan x} - x\right)} + \mathsf{fma}\left(-x, 1, x \cdot 1\right)} \]

      7. add-sqr-sqrt 95.0%

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

      8. *-commutative 95.0%

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

      9. *-un-lft-identity 95.0%

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

   8. Applied egg-rr 95.0%

      \[\leadsto \frac{-x}{\color{blue}{\left(\tan x - x\right) + \mathsf{fma}\left(-x, 1, x\right)}} \]
   9. Step-by-step derivation

      1. +-commutative 95.0%

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

      2. fma-udef 95.0%

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

      3. *-rgt-identity 95.0%

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

      4. +-commutative 95.0%

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

      5. associate-+l+ 95.0%

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

      6. sub-neg 95.0%

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

      7. associate-+l+ 95.0%

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

      8. +-commutative 95.0%

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

      9. sub-neg 95.0%

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

      10. unsub-neg 95.0%

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

   10. Simplified 95.0%

       \[\leadsto \frac{-x}{\color{blue}{x + \left(\left(\tan x - x\right) - x\right)}} \]
   11. Step-by-step derivation

       1. associate--l- 95.0%

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

       2. sub-neg 95.0%

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

   12. Applied egg-rr 95.0%

       \[\leadsto \frac{-x}{x + \color{blue}{\left(\tan x + \left(-\left(x + x\right)\right)\right)}} \]
   13. Step-by-step derivation

       1. unsub-neg 95.0%

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

   14. Simplified 95.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{x + \left(\tan x - \left(x + x\right)\right)}\\ \end{array} \]

Alternative 5: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.2) (+ (* 0.225 (* x x)) -0.5) (/ x (- x (tan x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = x / (x - tan(x));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.2d0) then
        tmp = (0.225d0 * (x * x)) + (-0.5d0)
    else
        tmp = x / (x - tan(x))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = x / (x - Math.tan(x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = (0.225 * (x * x)) + -0.5
	else:
		tmp = x / (x - math.tan(x))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64(Float64(0.225 * Float64(x * x)) + -0.5);
	else
		tmp = Float64(x / Float64(x - tan(x)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = (0.225 * (x * x)) + -0.5;
	else
		tmp = x / (x - tan(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.2], N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(x / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 34.1%

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

      1. sub-neg 34.1%

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

      2. +-commutative 34.1%

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

      3. neg-sub0 34.1%

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

      4. associate-+l- 34.1%

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

      5. sub0-neg 34.1%

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

      6. neg-mul-1 34.1%

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

      7. sub-neg 34.1%

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

      8. +-commutative 34.1%

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

      9. neg-sub0 34.1%

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

      10. associate-+l- 34.1%

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

      11. sub0-neg 34.1%

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

      12. neg-mul-1 34.1%

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

      13. times-frac 34.1%

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

      14. metadata-eval 34.1%

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

      15. *-lft-identity 34.1%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in x around 0 68.4%

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

      1. fma-neg 68.4%

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

      2. unpow2 68.4%

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

      3. metadata-eval 68.4%

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

   6. Simplified 68.4%

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

      1. fma-udef 68.4%

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

   8. Applied egg-rr 68.4%

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

   if 2.2000000000000002 < 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 95.0%

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

      1. neg-mul-1 95.0%

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

   6. Simplified 95.0%

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

      1. frac-2neg 95.0%

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

      2. div-inv 94.9%

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

      3. remove-double-neg 94.9%

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

      4. sub-neg 94.9%

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

      5. distribute-neg-in 94.9%

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

      6. remove-double-neg 94.9%

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

   8. Applied egg-rr 94.9%

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

      1. associate-*r/ 95.0%

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

      2. *-rgt-identity 95.0%

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

      3. +-commutative 95.0%

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

      4. unsub-neg 95.0%

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

   10. Simplified 95.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \end{array} \]

Alternative 6: 98.8% accurate, 22.8× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 2.6) (+ (* 0.225 (* x x)) -0.5) 1.0))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.6d0) then
        tmp = (0.225d0 * (x * x)) + (-0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.6], N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 34.1%

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

      1. sub-neg 34.1%

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

      2. +-commutative 34.1%

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

      3. neg-sub0 34.1%

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

      4. associate-+l- 34.1%

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

      5. sub0-neg 34.1%

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

      6. neg-mul-1 34.1%

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

      7. sub-neg 34.1%

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

      8. +-commutative 34.1%

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

      9. neg-sub0 34.1%

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

      10. associate-+l- 34.1%

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

      11. sub0-neg 34.1%

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

      12. neg-mul-1 34.1%

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

      13. times-frac 34.1%

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

      14. metadata-eval 34.1%

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

      15. *-lft-identity 34.1%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in x around 0 68.4%

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

      1. fma-neg 68.4%

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

      2. unpow2 68.4%

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

      3. metadata-eval 68.4%

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

   6. Simplified 68.4%

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

      1. fma-udef 68.4%

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

   8. Applied egg-rr 68.4%

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

   if 2.60000000000000009 < 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 95.0%

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

4. Final simplification 74.2%

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

Alternative 7: 98.5% accurate, 67.6× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.6) -0.5 1.0))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.6d0) then
        tmp = -0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.6], -0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 34.1%

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

      1. sub-neg 34.1%

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

      2. +-commutative 34.1%

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

      3. neg-sub0 34.1%

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

      4. associate-+l- 34.1%

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

      5. sub0-neg 34.1%

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

      6. neg-mul-1 34.1%

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

      7. sub-neg 34.1%

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

      8. +-commutative 34.1%

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

      9. neg-sub0 34.1%

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

      10. associate-+l- 34.1%

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

      11. sub0-neg 34.1%

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

      12. neg-mul-1 34.1%

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

      13. times-frac 34.1%

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

      14. metadata-eval 34.1%

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

      15. *-lft-identity 34.1%

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

   3. Simplified 34.1%

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

   4. Taylor expanded in x around 0 67.1%

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

   if 1.6000000000000001 < 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 95.0%

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

4. Final simplification 73.2%

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

Alternative 8: 50.4% accurate, 207.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ -0.5 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 -0.5)

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = -0.5d0
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return -0.5;
}

Python

x = abs(x)
def code(x):
	return -0.5

Julia

x = abs(x)
function code(x)
	return -0.5
end

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := -0.5

Derivation

1. Initial program 48.5%

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

   1. sub-neg 48.5%

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

   2. +-commutative 48.5%

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

   3. neg-sub0 48.5%

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

   4. associate-+l- 48.5%

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

   5. sub0-neg 48.5%

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

   6. neg-mul-1 48.5%

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

   7. sub-neg 48.5%

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

   8. +-commutative 48.5%

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

   9. neg-sub0 48.5%

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

   10. associate-+l- 48.5%

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

   11. sub0-neg 48.5%

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

   12. neg-mul-1 48.5%

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

   13. times-frac 48.5%

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

   14. metadata-eval 48.5%

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

   15. *-lft-identity 48.5%

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

3. Simplified 48.5%

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

4. Taylor expanded in x around 0 52.8%

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

5. Final simplification 52.8%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023238 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))