sintan (problem 3.4.5)

Percentage Accurate: 50.4% → 100.0%

Time: 25.5s

Alternatives: 12

Speedup: 207.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := x - \tan x\\ \mathbf{if}\;x \leq 0.102:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + \left(0.00024107142857142857 \cdot {x}^{6} + 0.225 \cdot {x}^{2}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t_0} - \frac{\sin x}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (- x (tan x))))
   (if (<= x 0.102)
     (-
      (+
       (* -0.009642857142857142 (pow x 4.0))
       (+ (* 0.00024107142857142857 (pow x 6.0)) (* 0.225 (pow x 2.0))))
      0.5)
     (- (/ x t_0) (/ (sin x) t_0)))))

C

x = abs(x);
double code(double x) {
	double t_0 = x - tan(x);
	double tmp;
	if (x <= 0.102) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + ((0.00024107142857142857 * pow(x, 6.0)) + (0.225 * pow(x, 2.0)))) - 0.5;
	} else {
		tmp = (x / t_0) - (sin(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 = x - tan(x)
    if (x <= 0.102d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + ((0.00024107142857142857d0 * (x ** 6.0d0)) + (0.225d0 * (x ** 2.0d0)))) - 0.5d0
    else
        tmp = (x / t_0) - (sin(x) / t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = x - Math.tan(x);
	double tmp;
	if (x <= 0.102) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + ((0.00024107142857142857 * Math.pow(x, 6.0)) + (0.225 * Math.pow(x, 2.0)))) - 0.5;
	} else {
		tmp = (x / t_0) - (Math.sin(x) / t_0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = x - math.tan(x)
	tmp = 0
	if x <= 0.102:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + ((0.00024107142857142857 * math.pow(x, 6.0)) + (0.225 * math.pow(x, 2.0)))) - 0.5
	else:
		tmp = (x / t_0) - (math.sin(x) / t_0)
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(x - tan(x))
	tmp = 0.0
	if (x <= 0.102)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(Float64(0.00024107142857142857 * (x ^ 6.0)) + Float64(0.225 * (x ^ 2.0)))) - 0.5);
	else
		tmp = Float64(Float64(x / t_0) - Float64(sin(x) / t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = x - tan(x);
	tmp = 0.0;
	if (x <= 0.102)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + ((0.00024107142857142857 * (x ^ 6.0)) + (0.225 * (x ^ 2.0)))) - 0.5;
	else
		tmp = (x / t_0) - (sin(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[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.102], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.00024107142857142857 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.225 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.101999999999999993

   1. Initial program 33.7%

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

   2. Taylor expanded in x around 0 67.8%

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

   if 0.101999999999999993 < x

   1. Initial program 99.9%

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

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.102:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + \left(0.00024107142857142857 \cdot {x}^{6} + 0.225 \cdot {x}^{2}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := x - \tan x\\ \mathbf{if}\;x \leq 0.026:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t_0} - \frac{\sin x}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (- x (tan x))))
   (if (<= x 0.026)
     (+ (+ (* -0.009642857142857142 (pow x 4.0)) (* 0.225 (* x x))) -0.5)
     (- (/ x t_0) (/ (sin x) t_0)))))

C

x = abs(x);
double code(double x) {
	double t_0 = x - tan(x);
	double tmp;
	if (x <= 0.026) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + (0.225 * (x * x))) + -0.5;
	} else {
		tmp = (x / t_0) - (sin(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 = x - tan(x)
    if (x <= 0.026d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + (0.225d0 * (x * x))) + (-0.5d0)
    else
        tmp = (x / t_0) - (sin(x) / t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = x - Math.tan(x);
	double tmp;
	if (x <= 0.026) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + (0.225 * (x * x))) + -0.5;
	} else {
		tmp = (x / t_0) - (Math.sin(x) / t_0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = x - math.tan(x)
	tmp = 0
	if x <= 0.026:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + (0.225 * (x * x))) + -0.5
	else:
		tmp = (x / t_0) - (math.sin(x) / t_0)
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(x - tan(x))
	tmp = 0.0
	if (x <= 0.026)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(0.225 * Float64(x * x))) + -0.5);
	else
		tmp = Float64(Float64(x / t_0) - Float64(sin(x) / t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = x - tan(x);
	tmp = 0.0;
	if (x <= 0.026)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + (0.225 * (x * x))) + -0.5;
	else
		tmp = (x / t_0) - (sin(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[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.026], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0259999999999999988

   1. Initial program 33.7%

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

   2. Taylor expanded in x around 0 67.6%

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

      1. sub-neg 67.6%

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

      2. fma-def 67.6%

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

      3. unpow2 67.6%

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

      4. metadata-eval 67.6%

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

   4. Simplified 67.6%

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

      1. fma-udef 67.6%

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

      2. +-commutative 67.6%

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

   6. Applied egg-rr 67.6%

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

   if 0.0259999999999999988 < x

   1. Initial program 99.9%

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

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.026:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x - \tan x}}{\frac{1}{x - \sin x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.03)
   (+ (+ (* -0.009642857142857142 (pow x 4.0)) (* 0.225 (* x x))) -0.5)
   (/ (/ 1.0 (- x (tan x))) (/ 1.0 (- x (sin x))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.03) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + (0.225 * (x * x))) + -0.5;
	} else {
		tmp = (1.0 / (x - tan(x))) / (1.0 / (x - sin(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 <= 0.03d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + (0.225d0 * (x * x))) + (-0.5d0)
    else
        tmp = (1.0d0 / (x - tan(x))) / (1.0d0 / (x - sin(x)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.03) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + (0.225 * (x * x))) + -0.5;
	} else {
		tmp = (1.0 / (x - Math.tan(x))) / (1.0 / (x - Math.sin(x)));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.03:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + (0.225 * (x * x))) + -0.5
	else:
		tmp = (1.0 / (x - math.tan(x))) / (1.0 / (x - math.sin(x)))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.03)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(0.225 * Float64(x * x))) + -0.5);
	else
		tmp = Float64(Float64(1.0 / Float64(x - tan(x))) / Float64(1.0 / Float64(x - sin(x))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.03)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + (0.225 * (x * x))) + -0.5;
	else
		tmp = (1.0 / (x - tan(x))) / (1.0 / (x - sin(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.03], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(N[(1.0 / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.029999999999999999

   1. Initial program 33.7%

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

   2. Taylor expanded in x around 0 67.6%

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

      1. sub-neg 67.6%

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

      2. fma-def 67.6%

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

      3. unpow2 67.6%

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

      4. metadata-eval 67.6%

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

   4. Simplified 67.6%

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

      1. fma-udef 67.6%

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

      2. +-commutative 67.6%

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

   6. Applied egg-rr 67.6%

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

   if 0.029999999999999999 < x

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. div-inv 99.8%

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

      3. associate-/r* 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x - \tan x}}{\frac{1}{x - \sin x}}\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.03)
   (+ (+ (* -0.009642857142857142 (pow x 4.0)) (* 0.225 (* x x))) -0.5)
   (/ 1.0 (/ (- x (tan x)) (- x (sin x))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.03) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + (0.225 * (x * x))) + -0.5;
	} else {
		tmp = 1.0 / ((x - tan(x)) / (x - sin(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 <= 0.03d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + (0.225d0 * (x * x))) + (-0.5d0)
    else
        tmp = 1.0d0 / ((x - tan(x)) / (x - sin(x)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.03) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + (0.225 * (x * x))) + -0.5;
	} else {
		tmp = 1.0 / ((x - Math.tan(x)) / (x - Math.sin(x)));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.03:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + (0.225 * (x * x))) + -0.5
	else:
		tmp = 1.0 / ((x - math.tan(x)) / (x - math.sin(x)))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.03)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(0.225 * Float64(x * x))) + -0.5);
	else
		tmp = Float64(1.0 / Float64(Float64(x - tan(x)) / Float64(x - sin(x))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.03)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + (0.225 * (x * x))) + -0.5;
	else
		tmp = 1.0 / ((x - tan(x)) / (x - sin(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.03], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(1.0 / N[(N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.029999999999999999

   1. Initial program 33.7%

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

   2. Taylor expanded in x around 0 67.6%

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

      1. sub-neg 67.6%

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

      2. fma-def 67.6%

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

      3. unpow2 67.6%

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

      4. metadata-eval 67.6%

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

   4. Simplified 67.6%

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

      1. fma-udef 67.6%

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

      2. +-commutative 67.6%

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

   6. Applied egg-rr 67.6%

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

   if 0.029999999999999999 < x

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \end{array} \]

Alternative 5: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.03)
   (+ (+ (* -0.009642857142857142 (pow x 4.0)) (* 0.225 (* x x))) -0.5)
   (/ (- x (sin x)) (- x (tan x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.03) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + (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 <= 0.03d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + (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 <= 0.03) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + (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 <= 0.03:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + (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 <= 0.03)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + 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 <= 0.03)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + (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, 0.03], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.225 * N[(x * x), $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 < 0.029999999999999999

   1. Initial program 33.7%

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

   2. Taylor expanded in x around 0 67.6%

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

      1. sub-neg 67.6%

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

      2. fma-def 67.6%

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

      3. unpow2 67.6%

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

      4. metadata-eval 67.6%

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

   4. Simplified 67.6%

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

      1. fma-udef 67.6%

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

      2. +-commutative 67.6%

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

   6. Applied egg-rr 67.6%

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

   if 0.029999999999999999 < x

   1. Initial program 99.9%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 6: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 3.0) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + (0.225 * (x * x))) + -0.5;
	} else {
		tmp = (x / (x - tan(x))) - (-3.0 / (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 <= 3.0d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + (0.225d0 * (x * x))) + (-0.5d0)
    else
        tmp = (x / (x - tan(x))) - ((-3.0d0) / (x * x))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 3

   1. Initial program 33.7%

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

   2. Taylor expanded in x around 0 67.6%

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

      1. sub-neg 67.6%

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

      2. fma-def 67.6%

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

      3. unpow2 67.6%

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

      4. metadata-eval 67.6%

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

   4. Simplified 67.6%

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

      1. fma-udef 67.6%

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

      2. +-commutative 67.6%

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

   6. Applied egg-rr 67.6%

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

   if 3 < x

   1. Initial program 99.9%

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

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 96.2%

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

      1. unpow2 96.2%

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

   6. Simplified 96.2%

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

4. Final simplification 74.5%

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

Alternative 7: 98.8% accurate, 1.8× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.8)
   (+ (* 0.225 (* x x)) -0.5)
   (- (/ x (- x (tan x))) (/ -3.0 (* x x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.8) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = (x / (x - tan(x))) - (-3.0 / (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 <= 2.8d0) then
        tmp = (0.225d0 * (x * x)) + (-0.5d0)
    else
        tmp = (x / (x - tan(x))) - ((-3.0d0) / (x * x))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.8], N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(N[(x / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-3.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7999999999999998

   1. Initial program 33.7%

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

   2. Taylor expanded in x around 0 68.5%

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

      1. fma-neg 68.5%

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

      2. unpow2 68.5%

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

      3. metadata-eval 68.5%

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

   4. Simplified 68.5%

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

      1. fma-udef 68.5%

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

   6. Applied egg-rr 68.5%

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

   if 2.7999999999999998 < x

   1. Initial program 99.9%

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

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 96.2%

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

      1. unpow2 96.2%

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

   6. Simplified 96.2%

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

4. Final simplification 75.2%

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

Alternative 8: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.3) (+ (* 0.225 (* x x)) -0.5) (/ x (- x (tan x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.3) {
		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 <= 2.3d0) 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 <= 2.3) {
		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 <= 2.3:
		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 <= 2.3)
		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 <= 2.3)
		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, 2.3], 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 < 2.2999999999999998

   1. Initial program 33.7%

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

   2. Taylor expanded in x around 0 68.5%

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

      1. fma-neg 68.5%

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

      2. unpow2 68.5%

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

      3. metadata-eval 68.5%

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

   4. Simplified 68.5%

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

      1. fma-udef 68.5%

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

   6. Applied egg-rr 68.5%

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

   if 2.2999999999999998 < x

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. div-inv 99.8%

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

      3. associate-/r* 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 96.2%

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

      1. expm1-log1p-u 96.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{x - \tan x}}{\frac{1}{x}}\right)\right)} \]

      2. expm1-udef 96.2%

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

      3. associate-/r/ 96.2%

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

      4. /-rgt-identity 96.2%

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

   8. Applied egg-rr 96.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{x - \tan x} \cdot x\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 96.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x - \tan x} \cdot x\right)\right)} \]

      2. expm1-log1p 96.0%

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

      3. associate-*l/ 96.2%

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

      4. *-lft-identity 96.2%

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

   10. Simplified 96.2%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \end{array} \]

Alternative 9: 98.8% accurate, 18.7× speedup

Math

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

FPCore

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

C

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

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.9) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = (x / x) - (-3.0 / (x * x));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.89999999999999991

   1. Initial program 33.7%

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

   2. Taylor expanded in x around 0 68.5%

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

      1. fma-neg 68.5%

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

      2. unpow2 68.5%

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

      3. metadata-eval 68.5%

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

   4. Simplified 68.5%

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

      1. fma-udef 68.5%

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

   6. Applied egg-rr 68.5%

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

   if 2.89999999999999991 < x

   1. Initial program 99.9%

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

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 96.2%

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

      1. unpow2 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in x around inf 96.1%

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

4. Final simplification 75.2%

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

Alternative 10: 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 33.7%

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

   2. Taylor expanded in x around 0 68.5%

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

      1. fma-neg 68.5%

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

      2. unpow2 68.5%

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

      3. metadata-eval 68.5%

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

   4. Simplified 68.5%

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

      1. fma-udef 68.5%

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

   6. Applied egg-rr 68.5%

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

   if 2.60000000000000009 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 96.1%

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

4. Final simplification 75.2%

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

Alternative 11: 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 33.7%

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

   2. 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. Taylor expanded in x around inf 96.1%

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

4. Final simplification 74.1%

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

Alternative 12: 50.7% 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 49.8%

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

2. Taylor expanded in x around 0 51.2%

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

3. Final simplification 51.2%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023280 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))