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

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 6

Speedup: 1.0×

• 24.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(1.0 - N[(N[(x * 0.12), $MachinePrecision] * x + N[(x * 0.253), $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. +-commutative 99.9%

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

   2. distribute-rgt-in 99.8%

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

   3. fma-def 99.9%

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

   4. *-commutative 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 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. Add Preprocessing

5. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   \[\leadsto 1 - x \cdot \left(x \cdot 0.12 + 0.253\right) \]
4. 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 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
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. div-sub 99.8%

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

   3. metadata-eval 99.8%

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

   4. *-commutative 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. metadata-eval 99.8%

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

   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%

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

4. Applied egg-rr 99.8%

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

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

6. Simplified 99.8%

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

7. Taylor expanded in x around inf 97.4%

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

   1. *-commutative 97.4%

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

9. Simplified 97.4%

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

10. Final simplification 97.4%

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

Alternative 5: 52.1% 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 50.2%

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

   1. *-commutative 50.2%

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

5. Simplified 50.2%

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

6. Final simplification 50.2%

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

Alternative 6: 50.3% 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. 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. div-sub 99.8%

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

   3. metadata-eval 99.8%

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

   4. *-commutative 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. metadata-eval 99.8%

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

   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%

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

4. Applied egg-rr 99.8%

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

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

6. Simplified 99.8%

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

7. Taylor expanded in x around inf 97.4%

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

   1. *-commutative 97.4%

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

9. Simplified 97.4%

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

10. Taylor expanded in x around 0 48.2%

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

11. Final simplification 48.2%

    \[\leadsto 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))