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

Percentage Accurate: 99.9% → 99.8%

Time: 6.6s

Alternatives: 6

Speedup: 1.0×

• 25.1% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 + N[(-0.253 * x), $MachinePrecision]), $MachinePrecision] + N[(-0.12 * N[Power[x, 2.0], $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. Step-by-step derivation

   1. associate--r+ 99.9%

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

   2. cancel-sign-sub-inv 99.9%

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

   3. *-commutative 99.9%

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

   4. cancel-sign-sub-inv 99.9%

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

   5. metadata-eval 99.9%

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

   6. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. fma-define 99.9%

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

   5. +-commutative 99.9%

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

   6. distribute-neg-in 99.9%

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

   7. distribute-rgt-neg-in 99.9%

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

   8. metadata-eval 99.9%

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

   9. metadata-eval 99.9%

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

   10. fma-define 99.9%

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

   11. metadata-eval 99.9%

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

   12. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

6. Final simplification 99.9%

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

Alternative 4: 97.9% 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 99.5%

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

4. Taylor expanded in x around inf 98.0%

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

   1. *-commutative 98.0%

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

6. Simplified 98.0%

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

Alternative 5: 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 99.8%

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

3. Taylor expanded in x around 0 50.8%

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

   1. *-commutative 50.8%

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

5. Simplified 50.8%

   \[\leadsto 1 - \color{blue}{x \cdot 0.253} \]
6. Add Preprocessing

Alternative 6: 49.5% 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 99.8%

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

3. Taylor expanded in x around 0 49.2%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))