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

Percentage Accurate: 99.9% → 99.8%

Time: 6.4s

Alternatives: 5

Speedup: 1.0×

• 24.1% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 - N[(N[(x * 0.253), $MachinePrecision] + N[(0.12 * N[Power[x, 2.0], $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. distribute-rgt-in 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

   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%

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

5. Final simplification 99.9%

   \[\leadsto 1 - \left(x \cdot 0.253 + 0.12 \cdot {x}^{2}\right) \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 - N[(x * N[(x * N[(0.12 + N[(0.253 / x), $MachinePrecision]), $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. 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 99.8%

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

   1. associate-*r/ 99.8%

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

   2. metadata-eval 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 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 4: 98.0% 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. 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 97.9%

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

   1. *-commutative 97.9%

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

7. Simplified 97.9%

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

8. Final simplification 97.9%

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

Alternative 5: 52.4% 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 99.8%

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

3. Taylor expanded in x around 0 50.3%

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

   1. *-commutative 50.3%

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

5. Simplified 50.3%

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

6. Final simplification 50.3%

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

Reproduce

herbie shell --seed 2024069 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))