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

Percentage Accurate: 99.9% → 99.8%

Time: 6.2s

Alternatives: 6

Speedup: 1.0×

• 25.8% 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 99.9%

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

   1. distribute-rgt-in 99.9%

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

   2. *-commutative 99.9%

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

   3. *-commutative 99.9%

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

   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, 0.1× 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 99.9%

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

   3. *-commutative 99.9%

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

   4. fma-define 99.9%

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

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 3: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. distribute-rgt-in 99.9%

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

   2. *-commutative 99.9%

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

   3. *-commutative 99.9%

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

   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. add-sqr-sqrt 99.8%

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

   2. sqrt-unprod 90.1%

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

   3. swap-sqr 90.1%

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

   4. metadata-eval 90.1%

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

   5. pow-prod-up 90.2%

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

   6. metadata-eval 90.2%

      \[\leadsto 1 - \left(x \cdot 0.253 + \sqrt{0.0144 \cdot {x}^{\color{blue}{4}}}\right) \]

6. Applied egg-rr 90.2%

   \[\leadsto 1 - \left(x \cdot 0.253 + \color{blue}{\sqrt{0.0144 \cdot {x}^{4}}}\right) \]
7. Step-by-step derivation

   1. sqrt-prod 90.1%

      \[\leadsto 1 - \left(x \cdot 0.253 + \color{blue}{\sqrt{0.0144} \cdot \sqrt{{x}^{4}}}\right) \]

   2. metadata-eval 90.1%

      \[\leadsto 1 - \left(x \cdot 0.253 + \color{blue}{0.12} \cdot \sqrt{{x}^{4}}\right) \]

   3. sqrt-pow1 99.9%

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

   4. metadata-eval 99.9%

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

   5. unpow2 99.9%

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

   6. associate-*l* 99.9%

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

   7. *-commutative 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 4: 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 5: 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 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
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. associate-*r/ 93.8%

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

   3. metadata-eval 93.8%

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

   4. swap-sqr 93.8%

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

   5. pow2 93.8%

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

   6. metadata-eval 93.8%

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

   7. *-commutative 93.8%

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

   8. cancel-sign-sub-inv 93.8%

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

   9. metadata-eval 93.8%

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

4. Applied egg-rr 93.8%

   \[\leadsto 1 - \color{blue}{\frac{x \cdot \left(0.064009 - {x}^{2} \cdot 0.0144\right)}{0.253 + -0.12 \cdot x}} \]
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 98.1%

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

   1. *-commutative 98.1%

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

9. Simplified 98.1%

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

10. Final simplification 98.1%

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

Alternative 6: 51.8% 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 49.2%

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

   1. *-commutative 49.2%

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

5. Simplified 49.2%

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

6. Final simplification 49.2%

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

Reproduce

herbie shell --seed 2024079 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))