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

Percentage Accurate: 99.9% → 99.9%

Time: 6.7s

Alternatives: 6

Speedup: 1.0×

• 25.7% 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 99.9%

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

   1. +-commutative 99.9%

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

   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 99.9%

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

2. Final simplification 99.9%

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

Alternative 3: 97.6% 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 99.9%

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

   1. flip-+ 99.9%

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

   2. associate-*r/ 91.9%

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

   3. metadata-eval 91.9%

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

   4. swap-sqr 91.9%

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

   5. pow2 91.9%

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

   6. metadata-eval 91.9%

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

   7. *-commutative 91.9%

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

   8. cancel-sign-sub-inv 91.9%

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

   9. metadata-eval 91.9%

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

3. Applied egg-rr 91.9%

   \[\leadsto 1 - \color{blue}{\frac{x \cdot \left(0.064009 - {x}^{2} \cdot 0.0144\right)}{0.253 + -0.12 \cdot x}} \]
4. Step-by-step derivation

   1. associate-/l* 99.8%

      \[\leadsto 1 - \color{blue}{\frac{x}{\frac{0.253 + -0.12 \cdot x}{0.064009 - {x}^{2} \cdot 0.0144}}} \]

   2. *-commutative 99.8%

      \[\leadsto 1 - \frac{x}{\frac{0.253 + \color{blue}{x \cdot -0.12}}{0.064009 - {x}^{2} \cdot 0.0144}} \]

5. Simplified 99.8%

   \[\leadsto 1 - \color{blue}{\frac{x}{\frac{0.253 + x \cdot -0.12}{0.064009 - {x}^{2} \cdot 0.0144}}} \]

6. Taylor expanded in x around inf 97.4%

   \[\leadsto 1 - \frac{x}{\color{blue}{\frac{8.333333333333334}{x}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 97.4%

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

   2. div-inv 97.3%

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

   3. times-frac 97.4%

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

   4. metadata-eval 97.4%

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

8. Applied egg-rr 97.4%

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

   1. pow1 97.4%

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

   2. inv-pow 97.4%

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

   3. pow-div 97.4%

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

   4. metadata-eval 97.4%

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

   5. unpow2 97.4%

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

10. Applied egg-rr 97.4%

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

11. Final simplification 97.4%

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

Alternative 4: 97.6% 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. Step-by-step derivation

   1. flip-+ 99.9%

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

   2. div-sub 99.9%

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

   3. metadata-eval 99.9%

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

   4. *-commutative 99.9%

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

   5. cancel-sign-sub-inv 99.9%

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

   6. metadata-eval 99.9%

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

   7. swap-sqr 99.8%

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

   8. pow2 99.8%

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

   9. metadata-eval 99.8%

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

   10. *-commutative 99.8%

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

   11. cancel-sign-sub-inv 99.8%

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

   12. metadata-eval 99.8%

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

3. Applied egg-rr 99.8%

   \[\leadsto 1 - x \cdot \color{blue}{\left(\frac{0.064009}{0.253 + -0.12 \cdot x} - \frac{{x}^{2} \cdot 0.0144}{0.253 + -0.12 \cdot x}\right)} \]
4. Step-by-step derivation

   1. div-sub 99.8%

      \[\leadsto 1 - x \cdot \color{blue}{\frac{0.064009 - {x}^{2} \cdot 0.0144}{0.253 + -0.12 \cdot x}} \]

   2. *-commutative 99.8%

      \[\leadsto 1 - x \cdot \frac{0.064009 - {x}^{2} \cdot 0.0144}{0.253 + \color{blue}{x \cdot -0.12}} \]

5. Simplified 99.8%

   \[\leadsto 1 - x \cdot \color{blue}{\frac{0.064009 - {x}^{2} \cdot 0.0144}{0.253 + x \cdot -0.12}} \]

6. Taylor expanded in x around inf 97.4%

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

7. Final simplification 97.4%

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

Alternative 5: 50.3% 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 99.9%

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

2. Taylor expanded in x around 0 53.5%

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

   1. *-commutative 53.5%

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

4. Simplified 53.5%

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

5. Final simplification 53.5%

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

Alternative 6: 48.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 51.4%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 51.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023308 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))