sintan (problem 3.4.5)

Percentage Accurate: 50.8% → 100.0%

Time: 19.0s

Alternatives: 7

Speedup: 67.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.8% 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, 0.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;\left(x \cdot \left(x \cdot 0.225\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.095)
   (-
    (+
     (* x (* x 0.225))
     (+
      (* -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 <= 0.095) {
		tmp = ((x * (x * 0.225)) + ((-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 <= 0.095d0) then
        tmp = ((x * (x * 0.225d0)) + (((-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 <= 0.095) {
		tmp = ((x * (x * 0.225)) + ((-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 <= 0.095:
		tmp = ((x * (x * 0.225)) + ((-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 <= 0.095)
		tmp = Float64(Float64(Float64(x * Float64(x * 0.225)) + 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 <= 0.095)
		tmp = ((x * (x * 0.225)) + ((-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, 0.095], N[(N[(N[(x * N[(x * 0.225), $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 < 0.095000000000000001

   1. Initial program 35.1%

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

   2. Taylor expanded in x around 0 67.6%

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

      1. add-log-exp 67.5%

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

      2. *-un-lft-identity 67.5%

         \[\leadsto \left(\log \color{blue}{\left(1 \cdot e^{0.225 \cdot {x}^{2}}\right)} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]

      3. log-prod 67.5%

         \[\leadsto \left(\color{blue}{\left(\log 1 + \log \left(e^{0.225 \cdot {x}^{2}}\right)\right)} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]

      4. metadata-eval 67.5%

         \[\leadsto \left(\left(\color{blue}{0} + \log \left(e^{0.225 \cdot {x}^{2}}\right)\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]

      5. add-log-exp 67.6%

         \[\leadsto \left(\left(0 + \color{blue}{0.225 \cdot {x}^{2}}\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]

      6. *-commutative 67.6%

         \[\leadsto \left(\left(0 + \color{blue}{{x}^{2} \cdot 0.225}\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]

      7. unpow2 67.6%

         \[\leadsto \left(\left(0 + \color{blue}{\left(x \cdot x\right)} \cdot 0.225\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]

      8. associate-*l* 67.6%

         \[\leadsto \left(\left(0 + \color{blue}{x \cdot \left(x \cdot 0.225\right)}\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]

   4. Applied egg-rr 67.6%

      \[\leadsto \left(\color{blue}{\left(0 + x \cdot \left(x \cdot 0.225\right)\right)} + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5 \]
   5. Step-by-step derivation

      1. +-lft-identity 67.6%

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

   6. Simplified 67.6%

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

   if 0.095000000000000001 < x

   1. Initial program 100.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;\left(x \cdot \left(x \cdot 0.225\right) + \left(-0.009642857142857142 \cdot {x}^{4} + 0.00024107142857142857 \cdot {x}^{6}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\sin x - \tan x}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.6)
   (+ (+ (* -0.009642857142857142 (pow x 4.0)) (* 0.225 (* x x))) -0.5)
   (- 1.0 (/ (- (sin x) (tan x)) x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + (0.225 * (x * x))) + -0.5;
	} else {
		tmp = 1.0 - ((sin(x) - tan(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 <= 2.6d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + (0.225d0 * (x * x))) + (-0.5d0)
    else
        tmp = 1.0d0 - ((sin(x) - tan(x)) / x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + (0.225 * (x * x))) + -0.5;
	} else {
		tmp = 1.0 - ((Math.sin(x) - Math.tan(x)) / x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.6:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + (0.225 * (x * x))) + -0.5
	else:
		tmp = 1.0 - ((math.sin(x) - math.tan(x)) / x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.6)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(0.225 * Float64(x * x))) + -0.5);
	else
		tmp = Float64(1.0 - Float64(Float64(sin(x) - tan(x)) / x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.6)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + (0.225 * (x * x))) + -0.5;
	else
		tmp = 1.0 - ((sin(x) - tan(x)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.6], 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[(N[Sin[x], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 35.1%

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

   2. Taylor expanded in x around 0 67.3%

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

      1. sub-neg 67.3%

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

      2. fma-def 67.3%

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

      3. unpow2 67.3%

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

      4. metadata-eval 67.3%

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

   4. Simplified 67.3%

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

      1. fma-udef 67.3%

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

   6. Applied egg-rr 67.3%

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

   if 2.60000000000000009 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. *-lft-identity 99.6%

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

      4. metadata-eval 99.6%

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

      5. cancel-sign-sub-inv 99.6%

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

      6. distribute-lft-out-- 99.6%

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

      7. mul-1-neg 99.6%

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

      8. remove-double-neg 99.6%

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

      9. associate-/r* 99.6%

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

      10. div-sub 99.6%

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

      11. mul-1-neg 99.6%

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

      12. unsub-neg 99.6%

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

   4. Simplified 99.6%

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

      1. tan-quot 99.6%

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

      2. sub-neg 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. sub-neg 99.6%

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

   8. Simplified 99.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\sin x - \tan x}{x}\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0235:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0235)
   (+ (+ (* -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 <= 0.0235) {
		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 <= 0.0235d0) 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 <= 0.0235) {
		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 <= 0.0235:
		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 <= 0.0235)
		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 <= 0.0235)
		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, 0.0235], 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 < 0.0235

   1. Initial program 34.8%

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

   2. Taylor expanded in x around 0 67.2%

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

      1. sub-neg 67.2%

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

      2. fma-def 67.2%

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

      3. unpow2 67.2%

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

      4. metadata-eval 67.2%

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

   4. Simplified 67.2%

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

      1. fma-udef 67.2%

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

   6. Applied egg-rr 67.2%

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

   if 0.0235 < x

   1. Initial program 99.8%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0235:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 4: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.9)
   (+ (+ (* -0.009642857142857142 (pow x 4.0)) (* 0.225 (* x x))) -0.5)
   1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.9) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + (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 <= 2.9d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + (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 <= 2.9) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + (0.225 * (x * x))) + -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.9:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + (0.225 * (x * x))) + -0.5
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.9)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + 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 <= 2.9)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + (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, 2.9], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.89999999999999991

   1. Initial program 35.1%

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

   2. Taylor expanded in x around 0 67.3%

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

      1. sub-neg 67.3%

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

      2. fma-def 67.3%

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

      3. unpow2 67.3%

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

      4. metadata-eval 67.3%

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

   4. Simplified 67.3%

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

      1. fma-udef 67.3%

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

   6. Applied egg-rr 67.3%

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

   if 2.89999999999999991 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 98.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot \left(x \cdot x\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 98.6% accurate, 22.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.55:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.55) (+ (* 0.225 (* x x)) -0.5) 1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.55) {
		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 <= 2.55d0) 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 <= 2.55) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.55:
		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 <= 2.55)
		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 <= 2.55)
		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, 2.55], N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.5499999999999998

   1. Initial program 35.1%

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

   2. Taylor expanded in x around 0 68.3%

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

      1. fma-neg 68.3%

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

      2. unpow2 68.3%

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

      3. metadata-eval 68.3%

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

   4. Simplified 68.3%

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

      1. fma-udef 68.3%

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

   6. Applied egg-rr 68.3%

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

   if 2.5499999999999998 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 98.9%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.55:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 98.3% 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 35.1%

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

   2. Taylor expanded in x around 0 66.9%

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

   if 1.55000000000000004 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 98.9%

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

4. Final simplification 76.4%

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

Alternative 7: 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 54.3%

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

2. Taylor expanded in x around 0 47.5%

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

3. Final simplification 47.5%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))