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

Percentage Accurate: 99.9% → 99.8%

Time: 7.2s

Alternatives: 5

Speedup: 1.0×

• 24.7% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 1 + {x}^{2} \cdot \left(\frac{-0.253}{x} - 0.12\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (* (pow x 2.0) (- (/ -0.253 x) 0.12))))

C

double code(double x) {
	return 1.0 + (pow(x, 2.0) * ((-0.253 / x) - 0.12));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((x ** 2.0d0) * (((-0.253d0) / x) - 0.12d0))
end function

Java

public static double code(double x) {
	return 1.0 + (Math.pow(x, 2.0) * ((-0.253 / x) - 0.12));
}

Python

def code(x):
	return 1.0 + (math.pow(x, 2.0) * ((-0.253 / x) - 0.12))

Julia

function code(x)
	return Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(-0.253 / x) - 0.12)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + ((x ^ 2.0) * ((-0.253 / x) - 0.12));
end

Wolfram

code[x_] := N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(-0.253 / x), $MachinePrecision] - 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 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. un-div-inv 99.9%

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

5. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

   2. remove-double-neg 99.9%

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

   3. unsub-neg 99.9%

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

   4. distribute-neg-frac 99.9%

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

   5. metadata-eval 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

   \[\leadsto 1 + {x}^{2} \cdot \left(\frac{-0.253}{x} - 0.12\right) \]
9. 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 3: 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. 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. clear-num 99.8%

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

   3. *-commutative 99.8%

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

   4. cancel-sign-sub-inv 99.8%

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

   5. metadata-eval 99.8%

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

   6. metadata-eval 99.8%

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

   7. swap-sqr 99.8%

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

   8. pow2 99.8%

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

   9. metadata-eval 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around inf 96.5%

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

   1. *-commutative 96.5%

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

7. Simplified 96.5%

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

8. Final simplification 96.5%

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

Alternative 4: 51.4% 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 inf 99.9%

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

   1. flip-+ 72.6%

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

   2. cancel-sign-sub-inv 72.6%

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

   3. metadata-eval 72.6%

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

   4. div-inv 72.6%

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

   5. associate-*r/ 72.6%

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

5. Applied egg-rr 69.7%

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

   1. *-commutative 69.7%

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

   2. associate-/l* 69.2%

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

7. Simplified 69.2%

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

8. Taylor expanded in x around 0 55.7%

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

9. Final simplification 55.7%

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

Alternative 5: 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 56.7%

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

   1. *-commutative 56.7%

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

5. Simplified 56.7%

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

6. Final simplification 56.7%

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

Reproduce

herbie shell --seed 2024054 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))