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

Percentage Accurate: 99.9% → 99.9%

Time: 8.0s

Alternatives: 5

Speedup: 1.0×

• 25.4% 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, 0.8× 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 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
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. *-un-lft-identity 99.9%

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

   2. +-commutative 99.9%

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

   3. un-div-inv 99.9%

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

5. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

   2. +-commutative 99.9%

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

   3. remove-double-neg 99.9%

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

   4. unsub-neg 99.9%

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

   5. distribute-neg-frac 99.9%

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

   6. metadata-eval 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 52.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;1 - \frac{x}{-3.952569169960474}\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot 0.253\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.2) (- 1.0 (/ x -3.952569169960474)) (- 1.0 (* x 0.253))))

C

double code(double x) {
	double tmp;
	if (x <= -3.2) {
		tmp = 1.0 - (x / -3.952569169960474);
	} else {
		tmp = 1.0 - (x * 0.253);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.2d0)) then
        tmp = 1.0d0 - (x / (-3.952569169960474d0))
    else
        tmp = 1.0d0 - (x * 0.253d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.2) {
		tmp = 1.0 - (x / -3.952569169960474);
	} else {
		tmp = 1.0 - (x * 0.253);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.2:
		tmp = 1.0 - (x / -3.952569169960474)
	else:
		tmp = 1.0 - (x * 0.253)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.2)
		tmp = Float64(1.0 - Float64(x / -3.952569169960474));
	else
		tmp = Float64(1.0 - Float64(x * 0.253));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.2)
		tmp = 1.0 - (x / -3.952569169960474);
	else
		tmp = 1.0 - (x * 0.253);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.2], N[(1.0 - N[(x / -3.952569169960474), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000002

   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 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. +-commutative 99.8%

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

      3. un-div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

      2. +-commutative 99.8%

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

      3. remove-double-neg 99.8%

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

      4. unsub-neg 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. 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%

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

   9. Applied egg-rr 99.3%

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

   10. Taylor expanded in x around 0 6.9%

       \[\leadsto 1 - \frac{x}{\color{blue}{-3.952569169960474}} \]

   if -3.2000000000000002 < x

   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 67.5%

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

      1. *-commutative 67.5%

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

   5. Simplified 67.5%

      \[\leadsto 1 - \color{blue}{x \cdot 0.253} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;1 - \frac{x}{-3.952569169960474}\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot 0.253\\ \end{array} \]
5. 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 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. 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. Taylor expanded in x around inf 97.9%

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

   1. *-commutative 97.9%

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

6. Simplified 97.9%

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

7. Final simplification 97.9%

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

Alternative 5: 50.8% 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 2024130 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))