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

Percentage Accurate: 99.9% → 99.7%

Time: 8.4s

Alternatives: 5

Speedup: 1.3×

• 24.4% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(x \cdot x, 0.0144, -0.064009\right)}{\mathsf{fma}\left(x, -0.12, 0.253\right)}, x, 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (/ (fma (* x x) 0.0144 -0.064009) (fma x -0.12 0.253)) x 1.0))

C

double code(double x) {
	return fma((fma((x * x), 0.0144, -0.064009) / fma(x, -0.12, 0.253)), x, 1.0);
}

Julia

function code(x)
	return fma(Float64(fma(Float64(x * x), 0.0144, -0.064009) / fma(x, -0.12, 0.253)), x, 1.0)
end

Wolfram

code[x_] := N[(N[(N[(N[(x * x), $MachinePrecision] * 0.0144 + -0.064009), $MachinePrecision] / N[(x * -0.12 + 0.253), $MachinePrecision]), $MachinePrecision] * x + 1.0), $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. sub-neg N/A

      \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right) + 1} \]

   3. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right) \cdot x}\right)\right) + 1 \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right) \cdot x} + 1 \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right), x, 1\right)} \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(x \cdot \frac{3}{25} + \frac{253}{1000}\right)}\right), x, 1\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{3}{25}\right)\right) + \left(\mathsf{neg}\left(\frac{253}{1000}\right)\right)}, x, 1\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{3}{25}\right)\right)} + \left(\mathsf{neg}\left(\frac{253}{1000}\right)\right), x, 1\right) \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(\frac{3}{25}\right), \mathsf{neg}\left(\frac{253}{1000}\right)\right)}, x, 1\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{\frac{-3}{25}}, \mathsf{neg}\left(\frac{253}{1000}\right)\right), x, 1\right) \]

   11. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, -0.12, -0.253\right), x, 1\right)} \]
5. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(x \cdot \frac{-3}{25}\right) \cdot \left(x \cdot \frac{-3}{25}\right) - \frac{-253}{1000} \cdot \frac{-253}{1000}}{x \cdot \frac{-3}{25} - \frac{-253}{1000}}}, x, 1\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(x \cdot \frac{-3}{25}\right) \cdot \left(x \cdot \frac{-3}{25}\right) - \frac{-253}{1000} \cdot \frac{-253}{1000}}{\color{blue}{x \cdot \frac{-3}{25} + \left(\mathsf{neg}\left(\frac{-253}{1000}\right)\right)}}, x, 1\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(x \cdot \frac{-3}{25}\right) \cdot \left(x \cdot \frac{-3}{25}\right) - \frac{-253}{1000} \cdot \frac{-253}{1000}}{x \cdot \frac{-3}{25} + \color{blue}{\frac{253}{1000}}}, x, 1\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\left(x \cdot \frac{-3}{25}\right) \cdot \left(x \cdot \frac{-3}{25}\right) - \frac{-253}{1000} \cdot \frac{-253}{1000}}{x \cdot \frac{-3}{25} + \frac{253}{1000}}}, x, 1\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(x \cdot \frac{-3}{25}\right) \cdot \left(x \cdot \frac{-3}{25}\right) - \color{blue}{\frac{64009}{1000000}}}{x \cdot \frac{-3}{25} + \frac{253}{1000}}, x, 1\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(x \cdot \frac{-3}{25}\right) \cdot \left(x \cdot \frac{-3}{25}\right) + \left(\mathsf{neg}\left(\frac{64009}{1000000}\right)\right)}}{x \cdot \frac{-3}{25} + \frac{253}{1000}}, x, 1\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(x \cdot \frac{-3}{25}\right) \cdot \left(x \cdot \frac{-3}{25}\right) + \color{blue}{\frac{-64009}{1000000}}}{x \cdot \frac{-3}{25} + \frac{253}{1000}}, x, 1\right) \]

   8. swap-sqr N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{-3}{25} \cdot \frac{-3}{25}\right)} + \frac{-64009}{1000000}}{x \cdot \frac{-3}{25} + \frac{253}{1000}}, x, 1\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \color{blue}{\frac{9}{625}} + \frac{-64009}{1000000}}{x \cdot \frac{-3}{25} + \frac{253}{1000}}, x, 1\right) \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{9}{625}, \frac{-64009}{1000000}\right)}}{x \cdot \frac{-3}{25} + \frac{253}{1000}}, x, 1\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{9}{625}, \frac{-64009}{1000000}\right)}{x \cdot \frac{-3}{25} + \frac{253}{1000}}, x, 1\right) \]

   12. accelerator-lowering-fma.f64 99.9

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

6. Applied egg-rr 99.9%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, 0.0144, -0.064009\right)}{\mathsf{fma}\left(x, -0.12, 0.253\right)}}, x, 1\right) \]
7. Add Preprocessing

Alternative 2: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

      \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right) + 1} \]

   3. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right) \cdot x}\right)\right) + 1 \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right) \cdot x} + 1 \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right), x, 1\right)} \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(x \cdot \frac{3}{25} + \frac{253}{1000}\right)}\right), x, 1\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{3}{25}\right)\right) + \left(\mathsf{neg}\left(\frac{253}{1000}\right)\right)}, x, 1\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{3}{25}\right)\right)} + \left(\mathsf{neg}\left(\frac{253}{1000}\right)\right), x, 1\right) \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(\frac{3}{25}\right), \mathsf{neg}\left(\frac{253}{1000}\right)\right)}, x, 1\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{\frac{-3}{25}}, \mathsf{neg}\left(\frac{253}{1000}\right)\right), x, 1\right) \]

   11. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 3: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * -0.12), $MachinePrecision] * x + 1.0), $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. sub-neg N/A

      \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right) + 1} \]

   3. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right) \cdot x}\right)\right) + 1 \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right) \cdot x} + 1 \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right), x, 1\right)} \]

   6. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(x \cdot \frac{3}{25} + \frac{253}{1000}\right)}\right), x, 1\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{3}{25}\right)\right) + \left(\mathsf{neg}\left(\frac{253}{1000}\right)\right)}, x, 1\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{3}{25}\right)\right)} + \left(\mathsf{neg}\left(\frac{253}{1000}\right)\right), x, 1\right) \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(\frac{3}{25}\right), \mathsf{neg}\left(\frac{253}{1000}\right)\right)}, x, 1\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{\frac{-3}{25}}, \mathsf{neg}\left(\frac{253}{1000}\right)\right), x, 1\right) \]

   11. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-3}{25} \cdot x}, x, 1\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \frac{-3}{25}}, x, 1\right) \]

   2. *-lowering-*.f64 98.5

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

7. Simplified 98.5%

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

Alternative 4: 51.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{1 + \frac{-253}{1000} \cdot x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{-253}{1000} \cdot x + 1} \]

   2. accelerator-lowering-fma.f64 46.8

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

5. Simplified 46.8%

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

Alternative 5: 50.0% accurate, 17.0× 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. Taylor expanded in x around 0

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

5. Simplified 45.2%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))