sintan (problem 3.4.5)

Percentage Accurate: 50.9% → 100.0%

Time: 16.1s

Alternatives: 9

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: 50.9% 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: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.086:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.086)
   (-
    (+
     (* 0.225 (pow x 2.0))
     (+
      (* -0.009642857142857142 (pow x 4.0))
      (* 0.00024107142857142857 (pow x 6.0))))
    0.5)
   (/ (- x (sin x)) (- x (tan x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.086) {
		tmp = ((0.225 * pow(x, 2.0)) + ((-0.009642857142857142 * pow(x, 4.0)) + (0.00024107142857142857 * pow(x, 6.0)))) - 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 <= 0.086d0) then
        tmp = ((0.225d0 * (x ** 2.0d0)) + (((-0.009642857142857142d0) * (x ** 4.0d0)) + (0.00024107142857142857d0 * (x ** 6.0d0)))) - 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 <= 0.086) {
		tmp = ((0.225 * Math.pow(x, 2.0)) + ((-0.009642857142857142 * Math.pow(x, 4.0)) + (0.00024107142857142857 * Math.pow(x, 6.0)))) - 0.5;
	} else {
		tmp = (x - Math.sin(x)) / (x - Math.tan(x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.086:
		tmp = ((0.225 * math.pow(x, 2.0)) + ((-0.009642857142857142 * math.pow(x, 4.0)) + (0.00024107142857142857 * math.pow(x, 6.0)))) - 0.5
	else:
		tmp = (x - math.sin(x)) / (x - math.tan(x))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.086)
		tmp = Float64(Float64(Float64(0.225 * (x ^ 2.0)) + Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(0.00024107142857142857 * (x ^ 6.0)))) - 0.5);
	else
		tmp = 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 <= 0.086)
		tmp = ((0.225 * (x ^ 2.0)) + ((-0.009642857142857142 * (x ^ 4.0)) + (0.00024107142857142857 * (x ^ 6.0)))) - 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, 0.086], N[(N[(N[(0.225 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.00024107142857142857 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], 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 < 0.085999999999999993

   1. Initial program 33.8%

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

      1. sub-neg 33.8%

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

      2. +-commutative 33.8%

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

      3. neg-sub0 33.8%

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

      4. associate-+l- 33.8%

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

      5. sub0-neg 33.8%

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

      6. neg-mul-1 33.8%

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

      7. sub-neg 33.8%

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

      8. +-commutative 33.8%

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

      9. neg-sub0 33.8%

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

      10. associate-+l- 33.8%

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

      11. sub0-neg 33.8%

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

      12. neg-mul-1 33.8%

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

      13. times-frac 33.8%

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

      14. metadata-eval 33.8%

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

      15. *-lft-identity 33.8%

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

   3. Simplified 33.8%

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

   4. Taylor expanded in x around 0 67.4%

      \[\leadsto \color{blue}{\left(0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5} \]

   if 0.085999999999999993 < x

   1. Initial program 99.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.086:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \sin x - x\\ \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + -0.5\right) + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan x}{t_0} - \frac{x}{t_0}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sin x) x)))
   (if (<= x 0.032)
     (+ (+ (* -0.009642857142857142 (pow x 4.0)) -0.5) (* 0.225 (* x x)))
     (/ 1.0 (- (/ (tan x) t_0) (/ x t_0))))))

C

x = abs(x);
double code(double x) {
	double t_0 = sin(x) - x;
	double tmp;
	if (x <= 0.032) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + -0.5) + (0.225 * (x * x));
	} else {
		tmp = 1.0 / ((tan(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 = sin(x) - x
    if (x <= 0.032d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + (-0.5d0)) + (0.225d0 * (x * x))
    else
        tmp = 1.0d0 / ((tan(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.sin(x) - x;
	double tmp;
	if (x <= 0.032) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + -0.5) + (0.225 * (x * x));
	} else {
		tmp = 1.0 / ((Math.tan(x) / t_0) - (x / t_0));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = math.sin(x) - x
	tmp = 0
	if x <= 0.032:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + -0.5) + (0.225 * (x * x))
	else:
		tmp = 1.0 / ((math.tan(x) / t_0) - (x / t_0))
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(sin(x) - x)
	tmp = 0.0
	if (x <= 0.032)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + -0.5) + Float64(0.225 * Float64(x * x)));
	else
		tmp = Float64(1.0 / Float64(Float64(tan(x) / t_0) - Float64(x / t_0)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = sin(x) - x;
	tmp = 0.0;
	if (x <= 0.032)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + -0.5) + (0.225 * (x * x));
	else
		tmp = 1.0 / ((tan(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[Sin[x], $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, 0.032], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[Tan[x], $MachinePrecision] / t$95$0), $MachinePrecision] - N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.032000000000000001

   1. Initial program 33.8%

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

      1. sub-neg 33.8%

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

      2. +-commutative 33.8%

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

      3. neg-sub0 33.8%

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

      4. associate-+l- 33.8%

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

      5. sub0-neg 33.8%

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

      6. neg-mul-1 33.8%

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

      7. sub-neg 33.8%

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

      8. +-commutative 33.8%

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

      9. neg-sub0 33.8%

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

      10. associate-+l- 33.8%

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

      11. sub0-neg 33.8%

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

      12. neg-mul-1 33.8%

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

      13. times-frac 33.8%

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

      14. metadata-eval 33.8%

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

      15. *-lft-identity 33.8%

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

   3. Simplified 33.8%

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

   4. Taylor expanded in x around 0 67.0%

      \[\leadsto \color{blue}{\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5} \]
   5. Step-by-step derivation

      1. associate--l+ 67.0%

         \[\leadsto \color{blue}{0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} - 0.5\right)} \]

      2. fma-def 67.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, {x}^{2}, -0.009642857142857142 \cdot {x}^{4} - 0.5\right)} \]

      3. unpow2 67.0%

         \[\leadsto \mathsf{fma}\left(0.225, \color{blue}{x \cdot x}, -0.009642857142857142 \cdot {x}^{4} - 0.5\right) \]

      4. fma-neg 67.0%

         \[\leadsto \mathsf{fma}\left(0.225, x \cdot x, \color{blue}{\mathsf{fma}\left(-0.009642857142857142, {x}^{4}, -0.5\right)}\right) \]

      5. metadata-eval 67.0%

         \[\leadsto \mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left(-0.009642857142857142, {x}^{4}, \color{blue}{-0.5}\right)\right) \]

   6. Simplified 67.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left(-0.009642857142857142, {x}^{4}, -0.5\right)\right)} \]

   7. Taylor expanded in x around 0 67.0%

      \[\leadsto \mathsf{fma}\left(0.225, x \cdot x, \color{blue}{-0.009642857142857142 \cdot {x}^{4} - 0.5}\right) \]
   8. Step-by-step derivation

      1. fma-udef 67.0%

         \[\leadsto \color{blue}{0.225 \cdot \left(x \cdot x\right) + \left(-0.009642857142857142 \cdot {x}^{4} - 0.5\right)} \]

      2. sub-neg 67.0%

         \[\leadsto 0.225 \cdot \left(x \cdot x\right) + \color{blue}{\left(-0.009642857142857142 \cdot {x}^{4} + \left(-0.5\right)\right)} \]

      3. metadata-eval 67.0%

         \[\leadsto 0.225 \cdot \left(x \cdot x\right) + \left(-0.009642857142857142 \cdot {x}^{4} + \color{blue}{-0.5}\right) \]

      4. associate-+l+ 67.0%

         \[\leadsto \color{blue}{\left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot {x}^{4}\right) + -0.5} \]

      5. +-commutative 67.0%

         \[\leadsto \color{blue}{-0.5 + \left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot {x}^{4}\right)} \]

      6. +-commutative 67.0%

         \[\leadsto -0.5 + \color{blue}{\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right)} \]

      7. associate-+r+ 67.0%

         \[\leadsto \color{blue}{\left(-0.5 + -0.009642857142857142 \cdot {x}^{4}\right) + 0.225 \cdot \left(x \cdot x\right)} \]

   9. Applied egg-rr 67.0%

      \[\leadsto \color{blue}{\left(-0.5 + -0.009642857142857142 \cdot {x}^{4}\right) + 0.225 \cdot \left(x \cdot x\right)} \]

   if 0.032000000000000001 < 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%

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

      1. sub-div 99.9%

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

      2. clear-num 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. div-sub 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + -0.5\right) + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan x}{\sin x - x} - \frac{x}{\sin x - x}}\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.028:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + -0.5\right) + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan x - x}{\sin x - x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.028)
   (+ (+ (* -0.009642857142857142 (pow x 4.0)) -0.5) (* 0.225 (* x x)))
   (/ 1.0 (/ (- (tan x) x) (- (sin x) x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.028) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + -0.5) + (0.225 * (x * x));
	} else {
		tmp = 1.0 / ((tan(x) - x) / (sin(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 <= 0.028d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + (-0.5d0)) + (0.225d0 * (x * x))
    else
        tmp = 1.0d0 / ((tan(x) - x) / (sin(x) - x))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.028) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + -0.5) + (0.225 * (x * x));
	} else {
		tmp = 1.0 / ((Math.tan(x) - x) / (Math.sin(x) - x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.028:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + -0.5) + (0.225 * (x * x))
	else:
		tmp = 1.0 / ((math.tan(x) - x) / (math.sin(x) - x))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.028)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + -0.5) + Float64(0.225 * Float64(x * x)));
	else
		tmp = Float64(1.0 / Float64(Float64(tan(x) - x) / Float64(sin(x) - x)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.028)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + -0.5) + (0.225 * (x * x));
	else
		tmp = 1.0 / ((tan(x) - x) / (sin(x) - x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.028], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0280000000000000006

   1. Initial program 33.8%

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

      1. sub-neg 33.8%

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

      2. +-commutative 33.8%

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

      3. neg-sub0 33.8%

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

      4. associate-+l- 33.8%

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

      5. sub0-neg 33.8%

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

      6. neg-mul-1 33.8%

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

      7. sub-neg 33.8%

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

      8. +-commutative 33.8%

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

      9. neg-sub0 33.8%

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

      10. associate-+l- 33.8%

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

      11. sub0-neg 33.8%

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

      12. neg-mul-1 33.8%

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

      13. times-frac 33.8%

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

      14. metadata-eval 33.8%

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

      15. *-lft-identity 33.8%

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

   3. Simplified 33.8%

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

   4. Taylor expanded in x around 0 67.0%

      \[\leadsto \color{blue}{\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5} \]
   5. Step-by-step derivation

      1. associate--l+ 67.0%

         \[\leadsto \color{blue}{0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} - 0.5\right)} \]

      2. fma-def 67.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, {x}^{2}, -0.009642857142857142 \cdot {x}^{4} - 0.5\right)} \]

      3. unpow2 67.0%

         \[\leadsto \mathsf{fma}\left(0.225, \color{blue}{x \cdot x}, -0.009642857142857142 \cdot {x}^{4} - 0.5\right) \]

      4. fma-neg 67.0%

         \[\leadsto \mathsf{fma}\left(0.225, x \cdot x, \color{blue}{\mathsf{fma}\left(-0.009642857142857142, {x}^{4}, -0.5\right)}\right) \]

      5. metadata-eval 67.0%

         \[\leadsto \mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left(-0.009642857142857142, {x}^{4}, \color{blue}{-0.5}\right)\right) \]

   6. Simplified 67.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left(-0.009642857142857142, {x}^{4}, -0.5\right)\right)} \]

   7. Taylor expanded in x around 0 67.0%

      \[\leadsto \mathsf{fma}\left(0.225, x \cdot x, \color{blue}{-0.009642857142857142 \cdot {x}^{4} - 0.5}\right) \]
   8. Step-by-step derivation

      1. fma-udef 67.0%

         \[\leadsto \color{blue}{0.225 \cdot \left(x \cdot x\right) + \left(-0.009642857142857142 \cdot {x}^{4} - 0.5\right)} \]

      2. sub-neg 67.0%

         \[\leadsto 0.225 \cdot \left(x \cdot x\right) + \color{blue}{\left(-0.009642857142857142 \cdot {x}^{4} + \left(-0.5\right)\right)} \]

      3. metadata-eval 67.0%

         \[\leadsto 0.225 \cdot \left(x \cdot x\right) + \left(-0.009642857142857142 \cdot {x}^{4} + \color{blue}{-0.5}\right) \]

      4. associate-+l+ 67.0%

         \[\leadsto \color{blue}{\left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot {x}^{4}\right) + -0.5} \]

      5. +-commutative 67.0%

         \[\leadsto \color{blue}{-0.5 + \left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot {x}^{4}\right)} \]

      6. +-commutative 67.0%

         \[\leadsto -0.5 + \color{blue}{\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right)} \]

      7. associate-+r+ 67.0%

         \[\leadsto \color{blue}{\left(-0.5 + -0.009642857142857142 \cdot {x}^{4}\right) + 0.225 \cdot \left(x \cdot x\right)} \]

   9. Applied egg-rr 67.0%

      \[\leadsto \color{blue}{\left(-0.5 + -0.009642857142857142 \cdot {x}^{4}\right) + 0.225 \cdot \left(x \cdot x\right)} \]

   if 0.0280000000000000006 < 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%

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

      1. sub-div 99.9%

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

      2. clear-num 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.028:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + -0.5\right) + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan x - x}{\sin x - x}}\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.028:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + -0.5\right) + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.028)
   (+ (+ (* -0.009642857142857142 (pow x 4.0)) -0.5) (* 0.225 (* x x)))
   (/ (- x (sin x)) (- x (tan x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.028) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + -0.5) + (0.225 * (x * x));
	} 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 <= 0.028d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + (-0.5d0)) + (0.225d0 * (x * x))
    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 <= 0.028) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + -0.5) + (0.225 * (x * x));
	} else {
		tmp = (x - Math.sin(x)) / (x - Math.tan(x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.028:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + -0.5) + (0.225 * (x * x))
	else:
		tmp = (x - math.sin(x)) / (x - math.tan(x))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.028)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + -0.5) + Float64(0.225 * Float64(x * x)));
	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 <= 0.028)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + -0.5) + (0.225 * (x * x));
	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, 0.028], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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 < 0.0280000000000000006

   1. Initial program 33.8%

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

      1. sub-neg 33.8%

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

      2. +-commutative 33.8%

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

      3. neg-sub0 33.8%

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

      4. associate-+l- 33.8%

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

      5. sub0-neg 33.8%

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

      6. neg-mul-1 33.8%

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

      7. sub-neg 33.8%

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

      8. +-commutative 33.8%

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

      9. neg-sub0 33.8%

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

      10. associate-+l- 33.8%

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

      11. sub0-neg 33.8%

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

      12. neg-mul-1 33.8%

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

      13. times-frac 33.8%

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

      14. metadata-eval 33.8%

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

      15. *-lft-identity 33.8%

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

   3. Simplified 33.8%

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

   4. Taylor expanded in x around 0 67.0%

      \[\leadsto \color{blue}{\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5} \]
   5. Step-by-step derivation

      1. associate--l+ 67.0%

         \[\leadsto \color{blue}{0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} - 0.5\right)} \]

      2. fma-def 67.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, {x}^{2}, -0.009642857142857142 \cdot {x}^{4} - 0.5\right)} \]

      3. unpow2 67.0%

         \[\leadsto \mathsf{fma}\left(0.225, \color{blue}{x \cdot x}, -0.009642857142857142 \cdot {x}^{4} - 0.5\right) \]

      4. fma-neg 67.0%

         \[\leadsto \mathsf{fma}\left(0.225, x \cdot x, \color{blue}{\mathsf{fma}\left(-0.009642857142857142, {x}^{4}, -0.5\right)}\right) \]

      5. metadata-eval 67.0%

         \[\leadsto \mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left(-0.009642857142857142, {x}^{4}, \color{blue}{-0.5}\right)\right) \]

   6. Simplified 67.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left(-0.009642857142857142, {x}^{4}, -0.5\right)\right)} \]

   7. Taylor expanded in x around 0 67.0%

      \[\leadsto \mathsf{fma}\left(0.225, x \cdot x, \color{blue}{-0.009642857142857142 \cdot {x}^{4} - 0.5}\right) \]
   8. Step-by-step derivation

      1. fma-udef 67.0%

         \[\leadsto \color{blue}{0.225 \cdot \left(x \cdot x\right) + \left(-0.009642857142857142 \cdot {x}^{4} - 0.5\right)} \]

      2. sub-neg 67.0%

         \[\leadsto 0.225 \cdot \left(x \cdot x\right) + \color{blue}{\left(-0.009642857142857142 \cdot {x}^{4} + \left(-0.5\right)\right)} \]

      3. metadata-eval 67.0%

         \[\leadsto 0.225 \cdot \left(x \cdot x\right) + \left(-0.009642857142857142 \cdot {x}^{4} + \color{blue}{-0.5}\right) \]

      4. associate-+l+ 67.0%

         \[\leadsto \color{blue}{\left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot {x}^{4}\right) + -0.5} \]

      5. +-commutative 67.0%

         \[\leadsto \color{blue}{-0.5 + \left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot {x}^{4}\right)} \]

      6. +-commutative 67.0%

         \[\leadsto -0.5 + \color{blue}{\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right)} \]

      7. associate-+r+ 67.0%

         \[\leadsto \color{blue}{\left(-0.5 + -0.009642857142857142 \cdot {x}^{4}\right) + 0.225 \cdot \left(x \cdot x\right)} \]

   9. Applied egg-rr 67.0%

      \[\leadsto \color{blue}{\left(-0.5 + -0.009642857142857142 \cdot {x}^{4}\right) + 0.225 \cdot \left(x \cdot x\right)} \]

   if 0.0280000000000000006 < x

   1. Initial program 99.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.028:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + -0.5\right) + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 5: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + -0.5\right) + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{3}{x \cdot x} - -1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 3.1)
   (+ (+ (* -0.009642857142857142 (pow x 4.0)) -0.5) (* 0.225 (* x x)))
   (- (/ 3.0 (* x x)) -1.0)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + -0.5) + (0.225 * (x * x));
	} else {
		tmp = (3.0 / (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 <= 3.1d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + (-0.5d0)) + (0.225d0 * (x * x))
    else
        tmp = (3.0d0 / (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 <= 3.1) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + -0.5) + (0.225 * (x * x));
	} else {
		tmp = (3.0 / (x * x)) - -1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 3.1:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + -0.5) + (0.225 * (x * x))
	else:
		tmp = (3.0 / (x * x)) - -1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 3.1)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + -0.5) + Float64(0.225 * Float64(x * x)));
	else
		tmp = Float64(Float64(3.0 / Float64(x * x)) - -1.0);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.1)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + -0.5) + (0.225 * (x * x));
	else
		tmp = (3.0 / (x * x)) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 3.1], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] + N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000009

   1. Initial program 33.8%

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

      1. sub-neg 33.8%

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

      2. +-commutative 33.8%

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

      3. neg-sub0 33.8%

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

      4. associate-+l- 33.8%

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

      5. sub0-neg 33.8%

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

      6. neg-mul-1 33.8%

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

      7. sub-neg 33.8%

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

      8. +-commutative 33.8%

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

      9. neg-sub0 33.8%

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

      10. associate-+l- 33.8%

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

      11. sub0-neg 33.8%

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

      12. neg-mul-1 33.8%

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

      13. times-frac 33.8%

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

      14. metadata-eval 33.8%

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

      15. *-lft-identity 33.8%

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

   3. Simplified 33.8%

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

   4. Taylor expanded in x around 0 67.0%

      \[\leadsto \color{blue}{\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5} \]
   5. Step-by-step derivation

      1. associate--l+ 67.0%

         \[\leadsto \color{blue}{0.225 \cdot {x}^{2} + \left(-0.009642857142857142 \cdot {x}^{4} - 0.5\right)} \]

      2. fma-def 67.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, {x}^{2}, -0.009642857142857142 \cdot {x}^{4} - 0.5\right)} \]

      3. unpow2 67.0%

         \[\leadsto \mathsf{fma}\left(0.225, \color{blue}{x \cdot x}, -0.009642857142857142 \cdot {x}^{4} - 0.5\right) \]

      4. fma-neg 67.0%

         \[\leadsto \mathsf{fma}\left(0.225, x \cdot x, \color{blue}{\mathsf{fma}\left(-0.009642857142857142, {x}^{4}, -0.5\right)}\right) \]

      5. metadata-eval 67.0%

         \[\leadsto \mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left(-0.009642857142857142, {x}^{4}, \color{blue}{-0.5}\right)\right) \]

   6. Simplified 67.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, \mathsf{fma}\left(-0.009642857142857142, {x}^{4}, -0.5\right)\right)} \]

   7. Taylor expanded in x around 0 67.0%

      \[\leadsto \mathsf{fma}\left(0.225, x \cdot x, \color{blue}{-0.009642857142857142 \cdot {x}^{4} - 0.5}\right) \]
   8. Step-by-step derivation

      1. fma-udef 67.0%

         \[\leadsto \color{blue}{0.225 \cdot \left(x \cdot x\right) + \left(-0.009642857142857142 \cdot {x}^{4} - 0.5\right)} \]

      2. sub-neg 67.0%

         \[\leadsto 0.225 \cdot \left(x \cdot x\right) + \color{blue}{\left(-0.009642857142857142 \cdot {x}^{4} + \left(-0.5\right)\right)} \]

      3. metadata-eval 67.0%

         \[\leadsto 0.225 \cdot \left(x \cdot x\right) + \left(-0.009642857142857142 \cdot {x}^{4} + \color{blue}{-0.5}\right) \]

      4. associate-+l+ 67.0%

         \[\leadsto \color{blue}{\left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot {x}^{4}\right) + -0.5} \]

      5. +-commutative 67.0%

         \[\leadsto \color{blue}{-0.5 + \left(0.225 \cdot \left(x \cdot x\right) + -0.009642857142857142 \cdot {x}^{4}\right)} \]

      6. +-commutative 67.0%

         \[\leadsto -0.5 + \color{blue}{\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right)} \]

      7. associate-+r+ 67.0%

         \[\leadsto \color{blue}{\left(-0.5 + -0.009642857142857142 \cdot {x}^{4}\right) + 0.225 \cdot \left(x \cdot x\right)} \]

   9. Applied egg-rr 67.0%

      \[\leadsto \color{blue}{\left(-0.5 + -0.009642857142857142 \cdot {x}^{4}\right) + 0.225 \cdot \left(x \cdot x\right)} \]

   if 3.10000000000000009 < 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%

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

   6. Taylor expanded in x around 0 95.0%

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

      1. unpow2 95.0%

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

   8. Simplified 95.0%

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

   9. Taylor expanded in x around inf 95.1%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + -0.5\right) + 0.225 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{3}{x \cdot x} - -1\\ \end{array} \]

Alternative 6: 98.7% accurate, 22.8× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 2.5) (+ -0.5 (* 0.225 (* x x))) 1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.5) {
		tmp = -0.5 + (0.225 * (x * x));
	} 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 <= 2.5d0) then
        tmp = (-0.5d0) + (0.225d0 * (x * x))
    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 <= 2.5) {
		tmp = -0.5 + (0.225 * (x * x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.5], N[(-0.5 + N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 33.8%

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

      1. sub-neg 33.8%

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

      2. +-commutative 33.8%

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

      3. neg-sub0 33.8%

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

      4. associate-+l- 33.8%

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

      5. sub0-neg 33.8%

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

      6. neg-mul-1 33.8%

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

      7. sub-neg 33.8%

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

      8. +-commutative 33.8%

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

      9. neg-sub0 33.8%

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

      10. associate-+l- 33.8%

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

      11. sub0-neg 33.8%

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

      12. neg-mul-1 33.8%

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

      13. times-frac 33.8%

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

      14. metadata-eval 33.8%

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

      15. *-lft-identity 33.8%

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

   3. Simplified 33.8%

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

   4. Taylor expanded in x around 0 68.0%

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

      1. fma-neg 68.0%

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

      2. unpow2 68.0%

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

      3. metadata-eval 68.0%

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

   6. Simplified 68.0%

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

      1. fma-udef 68.0%

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

   8. Applied egg-rr 68.0%

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

   if 2.5 < 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 95.1%

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

4. Final simplification 74.0%

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

Alternative 7: 98.7% accurate, 22.8× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.9) (+ -0.5 (* 0.225 (* x x))) (- (/ 3.0 (* x x)) -1.0)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.9) {
		tmp = -0.5 + (0.225 * (x * x));
	} else {
		tmp = (3.0 / (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 <= 2.9d0) then
        tmp = (-0.5d0) + (0.225d0 * (x * x))
    else
        tmp = (3.0d0 / (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 <= 2.9) {
		tmp = -0.5 + (0.225 * (x * x));
	} else {
		tmp = (3.0 / (x * x)) - -1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.9:
		tmp = -0.5 + (0.225 * (x * x))
	else:
		tmp = (3.0 / (x * x)) - -1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.9)
		tmp = Float64(-0.5 + Float64(0.225 * Float64(x * x)));
	else
		tmp = Float64(Float64(3.0 / Float64(x * x)) - -1.0);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.9)
		tmp = -0.5 + (0.225 * (x * x));
	else
		tmp = (3.0 / (x * x)) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.9], N[(-0.5 + N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.89999999999999991

   1. Initial program 33.8%

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

      1. sub-neg 33.8%

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

      2. +-commutative 33.8%

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

      3. neg-sub0 33.8%

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

      4. associate-+l- 33.8%

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

      5. sub0-neg 33.8%

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

      6. neg-mul-1 33.8%

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

      7. sub-neg 33.8%

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

      8. +-commutative 33.8%

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

      9. neg-sub0 33.8%

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

      10. associate-+l- 33.8%

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

      11. sub0-neg 33.8%

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

      12. neg-mul-1 33.8%

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

      13. times-frac 33.8%

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

      14. metadata-eval 33.8%

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

      15. *-lft-identity 33.8%

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

   3. Simplified 33.8%

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

   4. Taylor expanded in x around 0 68.0%

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

      1. fma-neg 68.0%

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

      2. unpow2 68.0%

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

      3. metadata-eval 68.0%

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

   6. Simplified 68.0%

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

      1. fma-udef 68.0%

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

   8. Applied egg-rr 68.0%

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

   if 2.89999999999999991 < 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%

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

   6. Taylor expanded in x around 0 95.0%

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

      1. unpow2 95.0%

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

   8. Simplified 95.0%

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

   9. Taylor expanded in x around inf 95.1%

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

4. Final simplification 74.0%

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

Alternative 8: 98.4% accurate, 67.6× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.55) -0.5 1.0))

C

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

Python

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

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.55)
		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 <= 1.55)
		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, 1.55], -0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 33.8%

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

      1. sub-neg 33.8%

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

      2. +-commutative 33.8%

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

      3. neg-sub0 33.8%

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

      4. associate-+l- 33.8%

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

      5. sub0-neg 33.8%

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

      6. neg-mul-1 33.8%

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

      7. sub-neg 33.8%

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

      8. +-commutative 33.8%

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

      9. neg-sub0 33.8%

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

      10. associate-+l- 33.8%

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

      11. sub0-neg 33.8%

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

      12. neg-mul-1 33.8%

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

      13. times-frac 33.8%

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

      14. metadata-eval 33.8%

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

      15. *-lft-identity 33.8%

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

   3. Simplified 33.8%

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

   4. Taylor expanded in x around 0 66.9%

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

   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 95.1%

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

4. Final simplification 73.0%

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

Alternative 9: 50.3% 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 48.3%

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

   1. sub-neg 48.3%

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

   2. +-commutative 48.3%

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

   3. neg-sub0 48.3%

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

   4. associate-+l- 48.3%

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

   5. sub0-neg 48.3%

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

   6. neg-mul-1 48.3%

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

   7. sub-neg 48.3%

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

   8. +-commutative 48.3%

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

   9. neg-sub0 48.3%

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

   10. associate-+l- 48.3%

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

   11. sub0-neg 48.3%

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

   12. neg-mul-1 48.3%

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

   13. times-frac 48.3%

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

   14. metadata-eval 48.3%

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

   15. *-lft-identity 48.3%

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

3. Simplified 48.3%

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

4. Taylor expanded in x around 0 52.6%

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

5. Final simplification 52.6%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023240 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))