sintan (problem 3.4.5)

Percentage Accurate: 50.3% → 99.9%

Time: 14.5s

Alternatives: 8

Speedup: 207.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := x - \sin x\\ \mathbf{if}\;x \leq 0.0043:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{t_0} - \frac{\tan x}{t_0}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (- x (sin x))))
   (if (<= x 0.0043)
     (+ (* 0.225 (* x x)) -0.5)
     (/ 1.0 (- (/ x t_0) (/ (tan x) t_0))))))

C

x = abs(x);
double code(double x) {
	double t_0 = x - sin(x);
	double tmp;
	if (x <= 0.0043) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = 1.0 / ((x / t_0) - (tan(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 - sin(x)
    if (x <= 0.0043d0) then
        tmp = (0.225d0 * (x * x)) + (-0.5d0)
    else
        tmp = 1.0d0 / ((x / t_0) - (tan(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.sin(x);
	double tmp;
	if (x <= 0.0043) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = 1.0 / ((x / t_0) - (Math.tan(x) / t_0));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = x - math.sin(x)
	tmp = 0
	if x <= 0.0043:
		tmp = (0.225 * (x * x)) + -0.5
	else:
		tmp = 1.0 / ((x / t_0) - (math.tan(x) / t_0))
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(x - sin(x))
	tmp = 0.0
	if (x <= 0.0043)
		tmp = Float64(Float64(0.225 * Float64(x * x)) + -0.5);
	else
		tmp = Float64(1.0 / Float64(Float64(x / t_0) - Float64(tan(x) / t_0)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = x - sin(x);
	tmp = 0.0;
	if (x <= 0.0043)
		tmp = (0.225 * (x * x)) + -0.5;
	else
		tmp = 1.0 / ((x / t_0) - (tan(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[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.0043], N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(1.0 / N[(N[(x / t$95$0), $MachinePrecision] - N[(N[Tan[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0043

   1. Initial program 37.2%

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

   2. Taylor expanded in x around 0 65.5%

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

      1. fma-neg 65.5%

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

      2. unpow2 65.5%

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

      3. metadata-eval 65.5%

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

   4. Simplified 65.5%

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

      1. fma-udef 65.5%

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

   6. Applied egg-rr 65.5%

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

   if 0.0043 < 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. Simplified 99.9%

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

      1. div-sub 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0043:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x}}\\ \end{array} \]

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0032:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0032)
   (+ (* 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 <= 0.0032) {
		tmp = (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 <= 0.0032d0) then
        tmp = (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 <= 0.0032) {
		tmp = (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 <= 0.0032:
		tmp = (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 <= 0.0032)
		tmp = Float64(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 <= 0.0032)
		tmp = (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, 0.0032], N[(N[(0.225 * N[(x * x), $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 < 0.00320000000000000015

   1. Initial program 37.2%

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

   2. Taylor expanded in x around 0 65.5%

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

      1. fma-neg 65.5%

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

      2. unpow2 65.5%

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

      3. metadata-eval 65.5%

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

   4. Simplified 65.5%

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

      1. fma-udef 65.5%

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

   6. Applied egg-rr 65.5%

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

   if 0.00320000000000000015 < 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. Simplified 99.9%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0032:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \end{array} \]

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.45:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.45) (+ (* 0.225 (* x x)) -0.5) (+ 1.0 (/ (- (tan x) (sin x)) x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.45) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = 1.0 + ((tan(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 <= 2.45d0) then
        tmp = (0.225d0 * (x * x)) + (-0.5d0)
    else
        tmp = 1.0d0 + ((tan(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 <= 2.45) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = 1.0 + ((Math.tan(x) - Math.sin(x)) / x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.45:
		tmp = (0.225 * (x * x)) + -0.5
	else:
		tmp = 1.0 + ((math.tan(x) - math.sin(x)) / x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.45)
		tmp = Float64(Float64(0.225 * Float64(x * x)) + -0.5);
	else
		tmp = Float64(1.0 + Float64(Float64(tan(x) - sin(x)) / x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.45)
		tmp = (0.225 * (x * x)) + -0.5;
	else
		tmp = 1.0 + ((tan(x) - sin(x)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.45], N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(1.0 + N[(N[(N[Tan[x], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4500000000000002

   1. Initial program 37.2%

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

   2. Taylor expanded in x around 0 65.5%

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

      1. fma-neg 65.5%

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

      2. unpow2 65.5%

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

      3. metadata-eval 65.5%

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

   4. Simplified 65.5%

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

      1. fma-udef 65.5%

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

   6. Applied egg-rr 65.5%

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

   if 2.4500000000000002 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 99.3%

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

      1. associate--l+ 99.3%

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

      2. sub-neg 99.3%

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

      3. *-lft-identity 99.3%

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

      4. metadata-eval 99.3%

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

      5. cancel-sign-sub-inv 99.3%

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

      6. distribute-lft-out-- 99.3%

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

      7. mul-1-neg 99.3%

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

      8. remove-double-neg 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-/r* 99.3%

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

      11. div-sub 99.3%

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

   4. Simplified 99.3%

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

      1. tan-quot 99.3%

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

      2. sub-neg 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. sub-neg 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.45:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0032:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0032) (+ (* 0.225 (* x x)) -0.5) (/ (- x (sin x)) (- x (tan x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0032) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = (x - sin(x)) / (x - tan(x));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0032d0) then
        tmp = (0.225d0 * (x * x)) + (-0.5d0)
    else
        tmp = (x - sin(x)) / (x - tan(x))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.0032) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = (x - Math.sin(x)) / (x - Math.tan(x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.0032:
		tmp = (0.225 * (x * x)) + -0.5
	else:
		tmp = (x - math.sin(x)) / (x - math.tan(x))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.0032)
		tmp = Float64(Float64(0.225 * Float64(x * x)) + -0.5);
	else
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0032)
		tmp = (0.225 * (x * x)) + -0.5;
	else
		tmp = (x - sin(x)) / (x - tan(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.0032], N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00320000000000000015

   1. Initial program 37.2%

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

   2. Taylor expanded in x around 0 65.5%

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

      1. fma-neg 65.5%

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

      2. unpow2 65.5%

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

      3. metadata-eval 65.5%

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

   4. Simplified 65.5%

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

      1. fma-udef 65.5%

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

   6. Applied egg-rr 65.5%

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

   if 0.00320000000000000015 < x

   1. Initial program 99.9%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0032:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 5: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.25) (+ (* 0.225 (* x x)) -0.5) (/ (- x (sin x)) x)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.25) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = (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 <= 2.25d0) then
        tmp = (0.225d0 * (x * x)) + (-0.5d0)
    else
        tmp = (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 <= 2.25) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = (x - Math.sin(x)) / x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.25:
		tmp = (0.225 * (x * x)) + -0.5
	else:
		tmp = (x - math.sin(x)) / x
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.25)
		tmp = Float64(Float64(0.225 * Float64(x * x)) + -0.5);
	else
		tmp = Float64(Float64(x - sin(x)) / x);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.25)
		tmp = (0.225 * (x * x)) + -0.5;
	else
		tmp = (x - sin(x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.25], N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 37.2%

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

   2. Taylor expanded in x around 0 65.5%

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

      1. fma-neg 65.5%

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

      2. unpow2 65.5%

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

      3. metadata-eval 65.5%

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

   4. Simplified 65.5%

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

      1. fma-udef 65.5%

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

   6. Applied egg-rr 65.5%

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

   if 2.25 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 96.7%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x}\\ \end{array} \]

Alternative 6: 98.8% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 37.2%

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

   2. Taylor expanded in x around 0 65.5%

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

      1. fma-neg 65.5%

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

      2. unpow2 65.5%

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

      3. metadata-eval 65.5%

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

   4. Simplified 65.5%

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

      1. fma-udef 65.5%

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

   6. Applied egg-rr 65.5%

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

   if 2.60000000000000009 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 96.6%

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

4. Final simplification 73.3%

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

Alternative 7: 98.5% accurate, 67.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.58:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.58) -0.5 1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.58) {
		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 <= 1.58d0) 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 <= 1.58) {
		tmp = -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.58:
		tmp = -0.5
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.58)
		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 <= 1.58)
		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, 1.58], -0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.5800000000000001

   1. Initial program 37.2%

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

   2. Taylor expanded in x around 0 64.3%

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

   if 1.5800000000000001 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 96.6%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.58:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 50.8% 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 52.9%

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

2. Taylor expanded in x around 0 48.6%

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

3. Final simplification 48.6%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))