sintan (problem 3.4.5)

Percentage Accurate: 49.6% → 100.0%

Time: 11.1s

Alternatives: 5

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: 49.6% 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, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot {x}^{2}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \end{array} \]

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 (pow x 2.0))) 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 * pow(x, 2.0))) - 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 <= 0.03d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + (0.225d0 * (x ** 2.0d0))) - 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 <= 0.03) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + (0.225 * Math.pow(x, 2.0))) - 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 <= 0.03:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + (0.225 * math.pow(x, 2.0))) - 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 <= 0.03)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(0.225 * (x ^ 2.0))) - 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 <= 0.03)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + (0.225 * (x ^ 2.0))) - 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, 0.03], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.225 * N[Power[x, 2.0], $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 27.6%

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

      1. /-rgt-identity 27.6%

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

      2. metadata-eval 27.6%

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

      3. associate-/r/ 27.6%

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

      4. metadata-eval 27.6%

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

      5. metadata-eval 27.6%

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

      6. times-frac 27.6%

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

      7. *-commutative 27.6%

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

      8. *-commutative 27.6%

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

      9. neg-mul-1 27.6%

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

      10. neg-sub0 27.6%

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

      11. associate-+l- 27.6%

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

      12. neg-sub0 27.6%

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

      13. +-commutative 27.6%

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

      14. unsub-neg 27.6%

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

      15. neg-mul-1 27.6%

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

      16. neg-sub0 27.6%

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

      17. associate-+l- 27.6%

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

      18. neg-sub0 27.6%

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

      19. +-commutative 27.6%

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

      20. unsub-neg 27.6%

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

   3. Simplified 27.6%

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

   4. Taylor expanded in x around 0 73.1%

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

   if 0.029999999999999999 < x

   1. Initial program 100.0%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot {x}^{2}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (- (* 0.225 (pow x 2.0)) 0.5)
   (/ (- x (sin x)) (- x (tan x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = (0.225 * pow(x, 2.0)) - 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 <= 0.0042d0) then
        tmp = (0.225d0 * (x ** 2.0d0)) - 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 <= 0.0042) {
		tmp = (0.225 * Math.pow(x, 2.0)) - 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 <= 0.0042:
		tmp = (0.225 * math.pow(x, 2.0)) - 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 <= 0.0042)
		tmp = Float64(Float64(0.225 * (x ^ 2.0)) - 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 <= 0.0042)
		tmp = (0.225 * (x ^ 2.0)) - 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, 0.0042], N[(N[(0.225 * N[Power[x, 2.0], $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 < 0.00419999999999999974

   1. Initial program 27.6%

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

      1. /-rgt-identity 27.6%

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

      2. metadata-eval 27.6%

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

      3. associate-/r/ 27.6%

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

      4. metadata-eval 27.6%

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

      5. metadata-eval 27.6%

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

      6. times-frac 27.6%

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

      7. *-commutative 27.6%

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

      8. *-commutative 27.6%

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

      9. neg-mul-1 27.6%

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

      10. neg-sub0 27.6%

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

      11. associate-+l- 27.6%

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

      12. neg-sub0 27.6%

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

      13. +-commutative 27.6%

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

      14. unsub-neg 27.6%

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

      15. neg-mul-1 27.6%

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

      16. neg-sub0 27.6%

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

      17. associate-+l- 27.6%

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

      18. neg-sub0 27.6%

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

      19. +-commutative 27.6%

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

      20. unsub-neg 27.6%

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

   3. Simplified 27.6%

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

   4. Taylor expanded in x around 0 74.0%

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - 0.5} \]

   if 0.00419999999999999974 < x

   1. Initial program 100.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 3: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.6) (- (* 0.225 (pow x 2.0)) 0.5) 1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = (0.225 * pow(x, 2.0)) - 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 ** 2.0d0)) - 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 * Math.pow(x, 2.0)) - 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.6:
		tmp = (0.225 * math.pow(x, 2.0)) - 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 * (x ^ 2.0)) - 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 ^ 2.0)) - 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[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 27.6%

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

      1. /-rgt-identity 27.6%

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

      2. metadata-eval 27.6%

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

      3. associate-/r/ 27.6%

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

      4. metadata-eval 27.6%

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

      5. metadata-eval 27.6%

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

      6. times-frac 27.6%

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

      7. *-commutative 27.6%

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

      8. *-commutative 27.6%

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

      9. neg-mul-1 27.6%

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

      10. neg-sub0 27.6%

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

      11. associate-+l- 27.6%

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

      12. neg-sub0 27.6%

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

      13. +-commutative 27.6%

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

      14. unsub-neg 27.6%

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

      15. neg-mul-1 27.6%

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

      16. neg-sub0 27.6%

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

      17. associate-+l- 27.6%

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

      18. neg-sub0 27.6%

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

      19. +-commutative 27.6%

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

      20. unsub-neg 27.6%

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

   3. Simplified 27.6%

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

   4. Taylor expanded in x around 0 74.0%

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - 0.5} \]

   if 2.60000000000000009 < x

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. times-frac 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. neg-mul-1 100.0%

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

      10. neg-sub0 100.0%

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

      11. associate-+l- 100.0%

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

      12. neg-sub0 100.0%

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

      13. +-commutative 100.0%

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

      14. unsub-neg 100.0%

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

      15. neg-mul-1 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate-+l- 100.0%

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

      18. neg-sub0 100.0%

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

      19. +-commutative 100.0%

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

      20. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.5%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 98.5% 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 27.6%

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

      1. /-rgt-identity 27.6%

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

      2. metadata-eval 27.6%

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

      3. associate-/r/ 27.6%

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

      4. metadata-eval 27.6%

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

      5. metadata-eval 27.6%

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

      6. times-frac 27.6%

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

      7. *-commutative 27.6%

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

      8. *-commutative 27.6%

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

      9. neg-mul-1 27.6%

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

      10. neg-sub0 27.6%

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

      11. associate-+l- 27.6%

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

      12. neg-sub0 27.6%

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

      13. +-commutative 27.6%

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

      14. unsub-neg 27.6%

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

      15. neg-mul-1 27.6%

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

      16. neg-sub0 27.6%

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

      17. associate-+l- 27.6%

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

      18. neg-sub0 27.6%

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

      19. +-commutative 27.6%

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

      20. unsub-neg 27.6%

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

   3. Simplified 27.6%

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

   4. Taylor expanded in x around 0 73.0%

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

   if 1.55000000000000004 < x

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. metadata-eval 100.0%

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

      6. times-frac 100.0%

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

      7. *-commutative 100.0%

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

      8. *-commutative 100.0%

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

      9. neg-mul-1 100.0%

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

      10. neg-sub0 100.0%

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

      11. associate-+l- 100.0%

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

      12. neg-sub0 100.0%

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

      13. +-commutative 100.0%

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

      14. unsub-neg 100.0%

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

      15. neg-mul-1 100.0%

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

      16. neg-sub0 100.0%

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

      17. associate-+l- 100.0%

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

      18. neg-sub0 100.0%

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

      19. +-commutative 100.0%

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

      20. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.5%

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

4. Final simplification 79.5%

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

Alternative 5: 51.5% 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 45.4%

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

   1. /-rgt-identity 45.4%

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

   2. metadata-eval 45.4%

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

   3. associate-/r/ 45.4%

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

   4. metadata-eval 45.4%

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

   5. metadata-eval 45.4%

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

   6. times-frac 45.4%

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

   7. *-commutative 45.4%

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

   8. *-commutative 45.4%

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

   9. neg-mul-1 45.4%

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

   10. neg-sub0 45.4%

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

   11. associate-+l- 45.4%

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

   12. neg-sub0 45.4%

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

   13. +-commutative 45.4%

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

   14. unsub-neg 45.4%

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

   15. neg-mul-1 45.4%

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

   16. neg-sub0 45.4%

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

   17. associate-+l- 45.4%

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

   18. neg-sub0 45.4%

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

   19. +-commutative 45.4%

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

   20. unsub-neg 45.4%

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

3. Simplified 45.4%

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

4. Taylor expanded in x around 0 55.4%

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

5. Final simplification 55.4%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023310 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))