Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A

Percentage Accurate: 99.9% → 99.9%

Time: 8.5s

Alternatives: 4

Speedup: 1.0×

• 24.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))

C

double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (0.253d0 + (x * 0.12d0)))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Python

def code(x):
	return 1.0 - (x * (0.253 + (x * 0.12)))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(0.253 + Float64(x * 0.12))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (0.253 + (x * 0.12)));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))

C

double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (0.253d0 + (x * 0.12d0)))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Python

def code(x):
	return 1.0 - (x * (0.253 + (x * 0.12)))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(0.253 + Float64(x * 0.12))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (0.253 + (x * 0.12)));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))

C

double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (0.253d0 + (x * 0.12d0)))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Python

def code(x):
	return 1.0 - (x * (0.253 + (x * 0.12)))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(0.253 + Float64(x * 0.12))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (0.253 + (x * 0.12)));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
4. Add Preprocessing

Alternative 2: 97.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot \frac{x}{8.333333333333334} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (/ x 8.333333333333334))))

C

double code(double x) {
	return 1.0 - (x * (x / 8.333333333333334));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (x / 8.333333333333334d0))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (x / 8.333333333333334));
}

Python

def code(x):
	return 1.0 - (x * (x / 8.333333333333334))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(x / 8.333333333333334)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (x / 8.333333333333334));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(x / 8.333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. distribute-rgt-in 99.9%

      \[\leadsto 1 - \color{blue}{\left(0.253 \cdot x + \left(x \cdot 0.12\right) \cdot x\right)} \]

   2. *-commutative 99.9%

      \[\leadsto 1 - \left(\color{blue}{x \cdot 0.253} + \left(x \cdot 0.12\right) \cdot x\right) \]

   3. *-commutative 99.9%

      \[\leadsto 1 - \left(x \cdot 0.253 + \color{blue}{\left(0.12 \cdot x\right)} \cdot x\right) \]

   4. associate-*l* 99.9%

      \[\leadsto 1 - \left(x \cdot 0.253 + \color{blue}{0.12 \cdot \left(x \cdot x\right)}\right) \]

   5. pow2 99.9%

      \[\leadsto 1 - \left(x \cdot 0.253 + 0.12 \cdot \color{blue}{{x}^{2}}\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto 1 - \color{blue}{\left(x \cdot 0.253 + 0.12 \cdot {x}^{2}\right)} \]

6. Applied egg-rr 99.9%

   \[\leadsto 1 - \color{blue}{\frac{1}{\frac{1}{x \cdot \mathsf{fma}\left(0.12, x, 0.253\right)}}} \]

7. Taylor expanded in x around inf 98.5%

   \[\leadsto 1 - \frac{1}{\frac{1}{x \cdot \color{blue}{\left(0.12 \cdot x\right)}}} \]
8. Step-by-step derivation

   1. *-commutative 98.5%

      \[\leadsto 1 - \frac{1}{\frac{1}{x \cdot \color{blue}{\left(x \cdot 0.12\right)}}} \]

9. Simplified 98.5%

   \[\leadsto 1 - \frac{1}{\frac{1}{x \cdot \color{blue}{\left(x \cdot 0.12\right)}}} \]
10. Step-by-step derivation

    1. remove-double-div 98.5%

       \[\leadsto 1 - \color{blue}{x \cdot \left(x \cdot 0.12\right)} \]

    2. associate-*r* 98.5%

       \[\leadsto 1 - \color{blue}{\left(x \cdot x\right) \cdot 0.12} \]

    3. unpow2 98.5%

       \[\leadsto 1 - \color{blue}{{x}^{2}} \cdot 0.12 \]

    4. metadata-eval 98.5%

       \[\leadsto 1 - {x}^{2} \cdot \color{blue}{\frac{1}{8.333333333333334}} \]

    5. div-inv 98.5%

       \[\leadsto 1 - \color{blue}{\frac{{x}^{2}}{8.333333333333334}} \]

    6. unpow2 98.5%

       \[\leadsto 1 - \frac{\color{blue}{x \cdot x}}{8.333333333333334} \]

    7. associate-/l* 98.5%

       \[\leadsto 1 - \color{blue}{x \cdot \frac{x}{8.333333333333334}} \]

11. Applied egg-rr 98.5%

    \[\leadsto 1 - \color{blue}{x \cdot \frac{x}{8.333333333333334}} \]

12. Final simplification 98.5%

    \[\leadsto 1 - x \cdot \frac{x}{8.333333333333334} \]
13. Add Preprocessing

Alternative 3: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(x \cdot 0.12\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (* x 0.12))))

C

double code(double x) {
	return 1.0 - (x * (x * 0.12));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (x * 0.12d0))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (x * 0.12));
}

Python

def code(x):
	return 1.0 - (x * (x * 0.12))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(x * 0.12)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (x * 0.12));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 98.5%

   \[\leadsto 1 - \color{blue}{0.12 \cdot {x}^{2}} \]
4. Step-by-step derivation

   1. add-cube-cbrt 98.2%

      \[\leadsto 1 - \color{blue}{\left(\sqrt[3]{0.12 \cdot {x}^{2}} \cdot \sqrt[3]{0.12 \cdot {x}^{2}}\right) \cdot \sqrt[3]{0.12 \cdot {x}^{2}}} \]

   2. pow3 98.2%

      \[\leadsto 1 - \color{blue}{{\left(\sqrt[3]{0.12 \cdot {x}^{2}}\right)}^{3}} \]

5. Applied egg-rr 98.2%

   \[\leadsto 1 - \color{blue}{{\left(\sqrt[3]{0.12 \cdot {x}^{2}}\right)}^{3}} \]
6. Step-by-step derivation

   1. rem-cube-cbrt 98.5%

      \[\leadsto 1 - \color{blue}{0.12 \cdot {x}^{2}} \]

   2. unpow2 98.5%

      \[\leadsto 1 - 0.12 \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. associate-*r* 98.5%

      \[\leadsto 1 - \color{blue}{\left(0.12 \cdot x\right) \cdot x} \]

   4. *-commutative 98.5%

      \[\leadsto 1 - \color{blue}{\left(x \cdot 0.12\right)} \cdot x \]

7. Applied egg-rr 98.5%

   \[\leadsto 1 - \color{blue}{\left(x \cdot 0.12\right) \cdot x} \]

8. Final simplification 98.5%

   \[\leadsto 1 - x \cdot \left(x \cdot 0.12\right) \]
9. Add Preprocessing

Alternative 4: 52.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot 0.253 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x 0.253)))

C

double code(double x) {
	return 1.0 - (x * 0.253);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * 0.253d0)
end function

Java

public static double code(double x) {
	return 1.0 - (x * 0.253);
}

Python

def code(x):
	return 1.0 - (x * 0.253)

Julia

function code(x)
	return Float64(1.0 - Float64(x * 0.253))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * 0.253);
end

Wolfram

code[x_] := N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 54.8%

   \[\leadsto 1 - \color{blue}{0.253 \cdot x} \]
4. Step-by-step derivation

   1. *-commutative 54.8%

      \[\leadsto 1 - \color{blue}{x \cdot 0.253} \]

5. Simplified 54.8%

   \[\leadsto 1 - \color{blue}{x \cdot 0.253} \]

6. Final simplification 54.8%

   \[\leadsto 1 - x \cdot 0.253 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))