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

Average Accuracy: 99.8% → 99.8%

Time: 8.2s

Precision: binary64

Cost: 704

Math

\[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

↓

\[1 - \frac{x}{\frac{1}{x \cdot 0.12 + 0.253}} \]

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 99.8%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

2. Applied egg-rr 83.5%

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

   [Start] 99.8	\[ 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   distribute-lft-in [=>] 99.8	\[ 1 - \color{blue}{\left(x \cdot 0.253 + x \cdot \left(x \cdot 0.12\right)\right)} \]
   flip-+ [=>] 83.5	\[ 1 - \color{blue}{\frac{\left(x \cdot 0.253\right) \cdot \left(x \cdot 0.253\right) - \left(x \cdot \left(x \cdot 0.12\right)\right) \cdot \left(x \cdot \left(x \cdot 0.12\right)\right)}{x \cdot 0.253 - x \cdot \left(x \cdot 0.12\right)}} \]

3. Simplified 99.8%

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

   [Start] 83.5	\[ 1 - \frac{\left(x \cdot 0.253\right) \cdot \left(x \cdot 0.253\right) - \left(x \cdot \left(x \cdot 0.12\right)\right) \cdot \left(x \cdot \left(x \cdot 0.12\right)\right)}{x \cdot 0.253 - x \cdot \left(x \cdot 0.12\right)} \]
   difference-of-squares [=>] 83.5	\[ 1 - \frac{\color{blue}{\left(x \cdot 0.253 + x \cdot \left(x \cdot 0.12\right)\right) \cdot \left(x \cdot 0.253 - x \cdot \left(x \cdot 0.12\right)\right)}}{x \cdot 0.253 - x \cdot \left(x \cdot 0.12\right)} \]
   +-commutative [=>] 83.5	\[ 1 - \frac{\color{blue}{\left(x \cdot \left(x \cdot 0.12\right) + x \cdot 0.253\right)} \cdot \left(x \cdot 0.253 - x \cdot \left(x \cdot 0.12\right)\right)}{x \cdot 0.253 - x \cdot \left(x \cdot 0.12\right)} \]
   distribute-lft-out [=>] 83.5	\[ 1 - \frac{\color{blue}{\left(x \cdot \left(x \cdot 0.12 + 0.253\right)\right)} \cdot \left(x \cdot 0.253 - x \cdot \left(x \cdot 0.12\right)\right)}{x \cdot 0.253 - x \cdot \left(x \cdot 0.12\right)} \]
   fma-udef [<=] 83.5	\[ 1 - \frac{\left(x \cdot \color{blue}{\mathsf{fma}\left(x, 0.12, 0.253\right)}\right) \cdot \left(x \cdot 0.253 - x \cdot \left(x \cdot 0.12\right)\right)}{x \cdot 0.253 - x \cdot \left(x \cdot 0.12\right)} \]
   associate-/l* [=>] 99.8	\[ 1 - \color{blue}{\frac{x \cdot \mathsf{fma}\left(x, 0.12, 0.253\right)}{\frac{x \cdot 0.253 - x \cdot \left(x \cdot 0.12\right)}{x \cdot 0.253 - x \cdot \left(x \cdot 0.12\right)}}} \]
   *-inverses [=>] 99.8	\[ 1 - \frac{x \cdot \mathsf{fma}\left(x, 0.12, 0.253\right)}{\color{blue}{1}} \]
   associate-/l* [=>] 99.8	\[ 1 - \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(x, 0.12, 0.253\right)}}} \]
   fma-udef [=>] 99.8	\[ 1 - \frac{x}{\frac{1}{\color{blue}{x \cdot 0.12 + 0.253}}} \]
   *-commutative [=>] 99.8	\[ 1 - \frac{x}{\frac{1}{\color{blue}{0.12 \cdot x} + 0.253}} \]
   fma-udef [<=] 99.8	\[ 1 - \frac{x}{\frac{1}{\color{blue}{\mathsf{fma}\left(0.12, x, 0.253\right)}}} \]

4. Applied egg-rr 99.8%

   \[\leadsto 1 - \frac{x}{\frac{1}{\color{blue}{x \cdot 0.12 + 0.253}}} \]
   Proof

   [Start] 99.8	\[ 1 - \frac{x}{\frac{1}{\mathsf{fma}\left(0.12, x, 0.253\right)}} \]
   fma-udef [=>] 99.8	\[ 1 - \frac{x}{\frac{1}{\color{blue}{0.12 \cdot x + 0.253}}} \]
   *-commutative [=>] 99.8	\[ 1 - \frac{x}{\frac{1}{\color{blue}{x \cdot 0.12} + 0.253}} \]

5. Final simplification 99.8%

   \[\leadsto 1 - \frac{x}{\frac{1}{x \cdot 0.12 + 0.253}} \]

Alternatives

Alternative 1
Accuracy	45.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \mathbf{else}:\\ \;\;\;\;1.5334083333333333\\ \end{array} \]

Alternative 2
Accuracy	97.9%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot 0.253\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	576

\[1 - x \cdot \left(x \cdot 0.12 + 0.253\right) \]

Alternative 4
Accuracy	97.2%
Cost	448

\[1 - 0.12 \cdot \left(x \cdot x\right) \]

Alternative 5
Accuracy	13.9%
Cost	64

\[1.5334083333333333 \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))