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

Percentage Accurate: 99.9% → 99.9%

Time: 8.5s

Alternatives: 5

Speedup: 1.0×

• 25.0% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. flip-+ 99.8%

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

   2. associate-*r/ 93.0%

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

   3. metadata-eval 93.0%

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

   4. swap-sqr 93.0%

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

   5. pow2 93.0%

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

   6. metadata-eval 93.0%

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

   7. *-commutative 93.0%

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

   8. cancel-sign-sub-inv 93.0%

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

   9. metadata-eval 93.0%

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

4. Applied egg-rr 93.0%

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

   1. associate-/l* 99.8%

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

   2. *-commutative 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in x around inf 99.9%

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

   1. associate-*r/ 99.9%

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

   2. metadata-eval 99.9%

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

   3. +-commutative 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

    \[\leadsto 1 - x \cdot \left(x \cdot \left(\frac{0.253}{x} + 0.12\right)\right) \]
11. Add Preprocessing

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. Add Preprocessing

3. Final simplification 99.8%

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

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 99.8%

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

3. Taylor expanded in x around inf 97.6%

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

   1. unpow2 97.6%

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

5. Applied egg-rr 97.6%

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

6. Final simplification 97.6%

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

Alternative 4: 97.7% 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. Add Preprocessing

3. Taylor expanded in x around inf 99.9%

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

4. Taylor expanded in x around inf 97.6%

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

   1. *-commutative 97.6%

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

6. Simplified 97.6%

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

7. Final simplification 97.6%

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

Alternative 5: 50.6% 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. Add Preprocessing

3. Taylor expanded in x around 0 55.6%

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

   1. *-commutative 55.6%

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

5. Simplified 55.6%

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

6. Final simplification 55.6%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))