ENA, Section 1.4, Exercise 4a

Percentage Accurate: 53.5% → 99.5%

Time: 25.3s

Alternatives: 6

Speedup: 2.0×

Specification

\[-1 \leq x \land x \leq 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left({x}^{2} \cdot -0.0007275132275132275 - 0.06388888888888888\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow x 2.0)
  (+
   0.16666666666666666
   (*
    (pow x 2.0)
    (- (* (pow x 2.0) -0.0007275132275132275) 0.06388888888888888)))))

C

double code(double x) {
	return pow(x, 2.0) * (0.16666666666666666 + (pow(x, 2.0) * ((pow(x, 2.0) * -0.0007275132275132275) - 0.06388888888888888)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 2.0d0) * (0.16666666666666666d0 + ((x ** 2.0d0) * (((x ** 2.0d0) * (-0.0007275132275132275d0)) - 0.06388888888888888d0)))
end function

Java

public static double code(double x) {
	return Math.pow(x, 2.0) * (0.16666666666666666 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * -0.0007275132275132275) - 0.06388888888888888)));
}

Python

def code(x):
	return math.pow(x, 2.0) * (0.16666666666666666 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * -0.0007275132275132275) - 0.06388888888888888)))

Julia

function code(x)
	return Float64((x ^ 2.0) * Float64(0.16666666666666666 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * -0.0007275132275132275) - 0.06388888888888888))))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ 2.0) * (0.16666666666666666 + ((x ^ 2.0) * (((x ^ 2.0) * -0.0007275132275132275) - 0.06388888888888888)));
end

Wolfram

code[x_] := N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.16666666666666666 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * -0.0007275132275132275), $MachinePrecision] - 0.06388888888888888), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 99.4%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(-0.0007275132275132275 \cdot {x}^{2} - 0.06388888888888888\right)\right)} \]

4. Final simplification 99.4%

   \[\leadsto {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left({x}^{2} \cdot -0.0007275132275132275 - 0.06388888888888888\right)\right) \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{2}}{\mathsf{fma}\left({x}^{2}, 2.3, 6\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (pow x 2.0) (fma (pow x 2.0) 2.3 6.0)))

C

double code(double x) {
	return pow(x, 2.0) / fma(pow(x, 2.0), 2.3, 6.0);
}

Julia

function code(x)
	return Float64((x ^ 2.0) / fma((x ^ 2.0), 2.3, 6.0))
end

Wolfram

code[x_] := N[(N[Power[x, 2.0], $MachinePrecision] / N[(N[Power[x, 2.0], $MachinePrecision] * 2.3 + 6.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. add-log-exp 54.7%

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

4. Applied egg-rr 54.7%

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

   1. rem-log-exp 54.6%

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

   2. clear-num 54.6%

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

6. Applied egg-rr 54.6%

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

7. Taylor expanded in x around 0 98.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{6 + 2.3 \cdot {x}^{2}}{{x}^{2}}}} \]
8. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{1}{\frac{6 + \color{blue}{{x}^{2} \cdot 2.3}}{{x}^{2}}} \]

9. Simplified 98.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{6 + {x}^{2} \cdot 2.3}{{x}^{2}}}} \]
10. Step-by-step derivation

    1. clear-num 99.2%

       \[\leadsto \color{blue}{\frac{{x}^{2}}{6 + {x}^{2} \cdot 2.3}} \]

    2. unpow2 99.2%

       \[\leadsto \frac{\color{blue}{x \cdot x}}{6 + {x}^{2} \cdot 2.3} \]

    3. associate-/l* 99.1%

       \[\leadsto \color{blue}{x \cdot \frac{x}{6 + {x}^{2} \cdot 2.3}} \]

    4. +-commutative 99.1%

       \[\leadsto x \cdot \frac{x}{\color{blue}{{x}^{2} \cdot 2.3 + 6}} \]

    5. fma-define 99.1%

       \[\leadsto x \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, 2.3, 6\right)}} \]

11. Applied egg-rr 99.1%

    \[\leadsto \color{blue}{x \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, 2.3, 6\right)}} \]
12. Step-by-step derivation

    1. associate-*r/ 99.2%

       \[\leadsto \color{blue}{\frac{x \cdot x}{\mathsf{fma}\left({x}^{2}, 2.3, 6\right)}} \]

    2. unpow2 99.2%

       \[\leadsto \frac{\color{blue}{{x}^{2}}}{\mathsf{fma}\left({x}^{2}, 2.3, 6\right)} \]

13. Simplified 99.2%

    \[\leadsto \color{blue}{\frac{{x}^{2}}{\mathsf{fma}\left({x}^{2}, 2.3, 6\right)}} \]

14. Final simplification 99.2%

    \[\leadsto \frac{{x}^{2}}{\mathsf{fma}\left({x}^{2}, 2.3, 6\right)} \]
15. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot -0.06388888888888888\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow x 2.0) (+ 0.16666666666666666 (* (pow x 2.0) -0.06388888888888888))))

C

double code(double x) {
	return pow(x, 2.0) * (0.16666666666666666 + (pow(x, 2.0) * -0.06388888888888888));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 2.0d0) * (0.16666666666666666d0 + ((x ** 2.0d0) * (-0.06388888888888888d0)))
end function

Java

public static double code(double x) {
	return Math.pow(x, 2.0) * (0.16666666666666666 + (Math.pow(x, 2.0) * -0.06388888888888888));
}

Python

def code(x):
	return math.pow(x, 2.0) * (0.16666666666666666 + (math.pow(x, 2.0) * -0.06388888888888888))

Julia

function code(x)
	return Float64((x ^ 2.0) * Float64(0.16666666666666666 + Float64((x ^ 2.0) * -0.06388888888888888)))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ 2.0) * (0.16666666666666666 + ((x ^ 2.0) * -0.06388888888888888));
end

Wolfram

code[x_] := N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.16666666666666666 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.06388888888888888), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 99.2%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.16666666666666666 + -0.06388888888888888 \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. *-commutative 99.2%

      \[\leadsto {x}^{2} \cdot \left(0.16666666666666666 + \color{blue}{{x}^{2} \cdot -0.06388888888888888}\right) \]

5. Simplified 99.2%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot -0.06388888888888888\right)} \]

6. Final simplification 99.2%

   \[\leadsto {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot -0.06388888888888888\right) \]
7. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, 2.3, 6\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (/ x (fma (pow x 2.0) 2.3 6.0))))

C

double code(double x) {
	return x * (x / fma(pow(x, 2.0), 2.3, 6.0));
}

Julia

function code(x)
	return Float64(x * Float64(x / fma((x ^ 2.0), 2.3, 6.0)))
end

Wolfram

code[x_] := N[(x * N[(x / N[(N[Power[x, 2.0], $MachinePrecision] * 2.3 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. add-log-exp 54.7%

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

4. Applied egg-rr 54.7%

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

   1. rem-log-exp 54.6%

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

   2. clear-num 54.6%

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

6. Applied egg-rr 54.6%

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

7. Taylor expanded in x around 0 98.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{6 + 2.3 \cdot {x}^{2}}{{x}^{2}}}} \]
8. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{1}{\frac{6 + \color{blue}{{x}^{2} \cdot 2.3}}{{x}^{2}}} \]

9. Simplified 98.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{6 + {x}^{2} \cdot 2.3}{{x}^{2}}}} \]
10. Step-by-step derivation

    1. clear-num 99.2%

       \[\leadsto \color{blue}{\frac{{x}^{2}}{6 + {x}^{2} \cdot 2.3}} \]

    2. unpow2 99.2%

       \[\leadsto \frac{\color{blue}{x \cdot x}}{6 + {x}^{2} \cdot 2.3} \]

    3. associate-/l* 99.1%

       \[\leadsto \color{blue}{x \cdot \frac{x}{6 + {x}^{2} \cdot 2.3}} \]

    4. +-commutative 99.1%

       \[\leadsto x \cdot \frac{x}{\color{blue}{{x}^{2} \cdot 2.3 + 6}} \]

    5. fma-define 99.1%

       \[\leadsto x \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, 2.3, 6\right)}} \]

11. Applied egg-rr 99.1%

    \[\leadsto \color{blue}{x \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, 2.3, 6\right)}} \]

12. Final simplification 99.1%

    \[\leadsto x \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, 2.3, 6\right)} \]
13. Add Preprocessing

Alternative 5: 98.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{2} \cdot 0.16666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow x 2.0) 0.16666666666666666))

C

double code(double x) {
	return pow(x, 2.0) * 0.16666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 2.0d0) * 0.16666666666666666d0
end function

Java

public static double code(double x) {
	return Math.pow(x, 2.0) * 0.16666666666666666;
}

Python

def code(x):
	return math.pow(x, 2.0) * 0.16666666666666666

Julia

function code(x)
	return Float64((x ^ 2.0) * 0.16666666666666666)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ 2.0) * 0.16666666666666666;
end

Wolfram

code[x_] := N[(N[Power[x, 2.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.4%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot {x}^{2}} \]

4. Final simplification 98.4%

   \[\leadsto {x}^{2} \cdot 0.16666666666666666 \]
5. Add Preprocessing

Alternative 6: 4.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.43478260869565216 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.43478260869565216)

C

double code(double x) {
	return 0.43478260869565216;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.43478260869565216d0
end function

Java

public static double code(double x) {
	return 0.43478260869565216;
}

Python

def code(x):
	return 0.43478260869565216

Julia

function code(x)
	return 0.43478260869565216
end

MATLAB

function tmp = code(x)
	tmp = 0.43478260869565216;
end

Wolfram

code[x_] := 0.43478260869565216

Derivation

1. Initial program 54.6%

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

   1. add-log-exp 54.7%

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

4. Applied egg-rr 54.7%

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

   1. rem-log-exp 54.6%

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

   2. clear-num 54.6%

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

6. Applied egg-rr 54.6%

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

7. Taylor expanded in x around 0 98.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{6 + 2.3 \cdot {x}^{2}}{{x}^{2}}}} \]
8. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{1}{\frac{6 + \color{blue}{{x}^{2} \cdot 2.3}}{{x}^{2}}} \]

9. Simplified 98.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{6 + {x}^{2} \cdot 2.3}{{x}^{2}}}} \]

10. Taylor expanded in x around inf 4.3%

    \[\leadsto \color{blue}{0.43478260869565216} \]

11. Final simplification 4.3%

    \[\leadsto 0.43478260869565216 \]
12. Add Preprocessing

Developer target: 98.8% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ 0.16666666666666666 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.16666666666666666 (* x x)))

C

double code(double x) {
	return 0.16666666666666666 * (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.16666666666666666d0 * (x * x)
end function

Java

public static double code(double x) {
	return 0.16666666666666666 * (x * x);
}

Python

def code(x):
	return 0.16666666666666666 * (x * x)

Julia

function code(x)
	return Float64(0.16666666666666666 * Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = 0.16666666666666666 * (x * x);
end

Wolfram

code[x_] := N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024100 
(FPCore (x)
  :name "ENA, Section 1.4, Exercise 4a"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))

  :alt
  (* 0.16666666666666666 (* x x))

  (/ (- x (sin x)) (tan x)))