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

Percentage Accurate: 99.9% → 99.9%

Time: 5.6s

Alternatives: 6

Speedup: 1.0×

• 25.2% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 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 99.8%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

2. Final simplification 99.8%

   \[\leadsto 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

Alternative 3: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

2. Taylor expanded in x around inf 97.5%

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

   1. unpow2 97.5%

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

4. Applied egg-rr 97.5%

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

5. Final simplification 97.5%

   \[\leadsto 1 - 0.12 \cdot \left(x \cdot x\right) \]

Alternative 4: 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 99.8%

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

   1. +-commutative 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. fma-def 99.8%

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

   4. *-commutative 99.8%

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

3. Applied egg-rr 99.8%

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

5. Applied egg-rr 99.8%

   \[\leadsto 1 - \color{blue}{\left(\frac{0.064009}{\frac{\mathsf{fma}\left(x, -0.12, 0.253\right)}{x}} - \frac{{\left(x \cdot 0.12\right)}^{2}}{\frac{\mathsf{fma}\left(x, -0.12, 0.253\right)}{x}}\right)} \]
6. Step-by-step derivation

   1. div-sub 99.8%

      \[\leadsto 1 - \color{blue}{\frac{0.064009 - {\left(x \cdot 0.12\right)}^{2}}{\frac{\mathsf{fma}\left(x, -0.12, 0.253\right)}{x}}} \]

   2. *-rgt-identity 99.8%

      \[\leadsto 1 - \frac{\color{blue}{\left(0.064009 - {\left(x \cdot 0.12\right)}^{2}\right) \cdot 1}}{\frac{\mathsf{fma}\left(x, -0.12, 0.253\right)}{x}} \]

   3. associate-*r/ 99.8%

      \[\leadsto 1 - \color{blue}{\left(0.064009 - {\left(x \cdot 0.12\right)}^{2}\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(x, -0.12, 0.253\right)}{x}}} \]

   4. associate-/r/ 99.7%

      \[\leadsto 1 - \left(0.064009 - {\left(x \cdot 0.12\right)}^{2}\right) \cdot \color{blue}{\left(\frac{1}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot x\right)} \]

   5. associate-*r* 99.7%

      \[\leadsto 1 - \color{blue}{\left(\left(0.064009 - {\left(x \cdot 0.12\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(x, -0.12, 0.253\right)}\right) \cdot x} \]

   6. associate-*r/ 99.8%

      \[\leadsto 1 - \color{blue}{\frac{\left(0.064009 - {\left(x \cdot 0.12\right)}^{2}\right) \cdot 1}{\mathsf{fma}\left(x, -0.12, 0.253\right)}} \cdot x \]

   7. *-rgt-identity 99.8%

      \[\leadsto 1 - \frac{\color{blue}{0.064009 - {\left(x \cdot 0.12\right)}^{2}}}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot x \]

   8. unpow2 99.8%

      \[\leadsto 1 - \frac{0.064009 - \color{blue}{\left(x \cdot 0.12\right) \cdot \left(x \cdot 0.12\right)}}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot x \]

   9. swap-sqr 99.7%

      \[\leadsto 1 - \frac{0.064009 - \color{blue}{\left(x \cdot x\right) \cdot \left(0.12 \cdot 0.12\right)}}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot x \]

   10. unpow2 99.7%

       \[\leadsto 1 - \frac{0.064009 - \color{blue}{{x}^{2}} \cdot \left(0.12 \cdot 0.12\right)}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot x \]

   11. metadata-eval 99.7%

       \[\leadsto 1 - \frac{0.064009 - {x}^{2} \cdot \color{blue}{0.0144}}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot x \]

7. Simplified 99.7%

   \[\leadsto 1 - \color{blue}{\frac{0.064009 - {x}^{2} \cdot 0.0144}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot x} \]

8. Taylor expanded in x around inf 97.5%

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

9. Final simplification 97.5%

   \[\leadsto 1 - x \cdot \left(x \cdot 0.12\right) \]

Alternative 5: 51.3% 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 99.8%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

2. Taylor expanded in x around 0 50.6%

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

   1. *-commutative 50.6%

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

4. Simplified 50.6%

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

5. Final simplification 50.6%

   \[\leadsto 1 - x \cdot 0.253 \]

Alternative 6: 10.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 1.5334083333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.5334083333333333)

C

double code(double x) {
	return 1.5334083333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.5334083333333333d0
end function

Java

public static double code(double x) {
	return 1.5334083333333333;
}

Python

def code(x):
	return 1.5334083333333333

Julia

function code(x)
	return 1.5334083333333333
end

MATLAB

function tmp = code(x)
	tmp = 1.5334083333333333;
end

Wolfram

code[x_] := 1.5334083333333333

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. fma-def 99.8%

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

   4. *-commutative 99.8%

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

3. Applied egg-rr 99.8%

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

5. Applied egg-rr 99.8%

   \[\leadsto 1 - \color{blue}{\left(\frac{0.064009}{\frac{\mathsf{fma}\left(x, -0.12, 0.253\right)}{x}} - \frac{{\left(x \cdot 0.12\right)}^{2}}{\frac{\mathsf{fma}\left(x, -0.12, 0.253\right)}{x}}\right)} \]
6. Step-by-step derivation

   1. div-sub 99.8%

      \[\leadsto 1 - \color{blue}{\frac{0.064009 - {\left(x \cdot 0.12\right)}^{2}}{\frac{\mathsf{fma}\left(x, -0.12, 0.253\right)}{x}}} \]

   2. *-rgt-identity 99.8%

      \[\leadsto 1 - \frac{\color{blue}{\left(0.064009 - {\left(x \cdot 0.12\right)}^{2}\right) \cdot 1}}{\frac{\mathsf{fma}\left(x, -0.12, 0.253\right)}{x}} \]

   3. associate-*r/ 99.8%

      \[\leadsto 1 - \color{blue}{\left(0.064009 - {\left(x \cdot 0.12\right)}^{2}\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(x, -0.12, 0.253\right)}{x}}} \]

   4. associate-/r/ 99.7%

      \[\leadsto 1 - \left(0.064009 - {\left(x \cdot 0.12\right)}^{2}\right) \cdot \color{blue}{\left(\frac{1}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot x\right)} \]

   5. associate-*r* 99.7%

      \[\leadsto 1 - \color{blue}{\left(\left(0.064009 - {\left(x \cdot 0.12\right)}^{2}\right) \cdot \frac{1}{\mathsf{fma}\left(x, -0.12, 0.253\right)}\right) \cdot x} \]

   6. associate-*r/ 99.8%

      \[\leadsto 1 - \color{blue}{\frac{\left(0.064009 - {\left(x \cdot 0.12\right)}^{2}\right) \cdot 1}{\mathsf{fma}\left(x, -0.12, 0.253\right)}} \cdot x \]

   7. *-rgt-identity 99.8%

      \[\leadsto 1 - \frac{\color{blue}{0.064009 - {\left(x \cdot 0.12\right)}^{2}}}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot x \]

   8. unpow2 99.8%

      \[\leadsto 1 - \frac{0.064009 - \color{blue}{\left(x \cdot 0.12\right) \cdot \left(x \cdot 0.12\right)}}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot x \]

   9. swap-sqr 99.7%

      \[\leadsto 1 - \frac{0.064009 - \color{blue}{\left(x \cdot x\right) \cdot \left(0.12 \cdot 0.12\right)}}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot x \]

   10. unpow2 99.7%

       \[\leadsto 1 - \frac{0.064009 - \color{blue}{{x}^{2}} \cdot \left(0.12 \cdot 0.12\right)}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot x \]

   11. metadata-eval 99.7%

       \[\leadsto 1 - \frac{0.064009 - {x}^{2} \cdot \color{blue}{0.0144}}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot x \]

7. Simplified 99.7%

   \[\leadsto 1 - \color{blue}{\frac{0.064009 - {x}^{2} \cdot 0.0144}{\mathsf{fma}\left(x, -0.12, 0.253\right)} \cdot x} \]

8. Taylor expanded in x around inf 59.0%

   \[\leadsto 1 - \frac{0.064009 - {x}^{2} \cdot 0.0144}{\color{blue}{-0.12 \cdot x}} \cdot x \]
9. Step-by-step derivation

   1. *-commutative 59.0%

      \[\leadsto 1 - \frac{0.064009 - {x}^{2} \cdot 0.0144}{\color{blue}{x \cdot -0.12}} \cdot x \]

10. Simplified 59.0%

    \[\leadsto 1 - \frac{0.064009 - {x}^{2} \cdot 0.0144}{\color{blue}{x \cdot -0.12}} \cdot x \]

11. Taylor expanded in x around 0 10.4%

    \[\leadsto \color{blue}{1.5334083333333333} \]

12. Final simplification 10.4%

    \[\leadsto 1.5334083333333333 \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))