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

Percentage Accurate: 99.9% → 99.8%

Time: 7.3s

Alternatives: 8

Speedup: 1.0×

• 24.1% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
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/ 93.0%

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

   3. metadata-eval 93.0%

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

   4. swap-sqr 93.0%

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

   5. pow2 93.0%

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

   6. metadata-eval 93.0%

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

   7. *-commutative 93.0%

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

   8. cancel-sign-sub-inv 93.0%

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

   9. metadata-eval 93.0%

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

4. Applied egg-rr 93.0%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.8%

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

   3. clear-num 99.8%

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

   4. metadata-eval 99.8%

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

   5. *-commutative 99.8%

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

   6. metadata-eval 99.8%

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

   7. unpow2 99.8%

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

   8. swap-sqr 99.8%

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

   9. *-commutative 99.8%

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

   10. flip-- 99.8%

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

   11. cancel-sign-sub-inv 99.8%

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

   12. metadata-eval 99.8%

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

   13. *-commutative 99.8%

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

   14. +-commutative 99.8%

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

   15. fma-define 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{x}{\frac{1}{0.253 + x \cdot 0.12}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
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/ 93.0%

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

   3. metadata-eval 93.0%

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

   4. swap-sqr 93.0%

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

   5. pow2 93.0%

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

   6. metadata-eval 93.0%

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

   7. *-commutative 93.0%

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

   8. cancel-sign-sub-inv 93.0%

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

   9. metadata-eval 93.0%

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

4. Applied egg-rr 93.0%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.8%

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

   3. clear-num 99.8%

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

   4. metadata-eval 99.8%

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

   5. *-commutative 99.8%

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

   6. metadata-eval 99.8%

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

   7. unpow2 99.8%

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

   8. swap-sqr 99.8%

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

   9. *-commutative 99.8%

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

   10. flip-- 99.8%

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

   11. cancel-sign-sub-inv 99.8%

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

   12. metadata-eval 99.8%

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

   13. *-commutative 99.8%

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

   14. +-commutative 99.8%

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

   15. fma-define 99.9%

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

8. Applied egg-rr 99.9%

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

   1. fma-undefine 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

    \[\leadsto 1 - \frac{x}{\frac{1}{0.253 + x \cdot 0.12}} \]
12. Add Preprocessing

Alternative 3: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - 0.12 \cdot \frac{x}{\frac{1}{x}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
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/ 93.0%

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

   3. metadata-eval 93.0%

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

   4. swap-sqr 93.0%

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

   5. pow2 93.0%

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

   6. metadata-eval 93.0%

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

   7. *-commutative 93.0%

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

   8. cancel-sign-sub-inv 93.0%

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

   9. metadata-eval 93.0%

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

4. Applied egg-rr 93.0%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.8%

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

   3. clear-num 99.8%

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

   4. metadata-eval 99.8%

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

   5. *-commutative 99.8%

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

   6. metadata-eval 99.8%

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

   7. unpow2 99.8%

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

   8. swap-sqr 99.8%

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

   9. *-commutative 99.8%

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

   10. flip-- 99.8%

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

   11. cancel-sign-sub-inv 99.8%

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

   12. metadata-eval 99.8%

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

   13. *-commutative 99.8%

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

   14. +-commutative 99.8%

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

   15. fma-define 99.9%

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

8. Applied egg-rr 99.9%

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

9. Taylor expanded in x around inf 96.9%

   \[\leadsto 1 - \frac{x}{\color{blue}{\frac{8.333333333333334}{x}}} \]
10. Step-by-step derivation

    1. *-un-lft-identity 96.9%

       \[\leadsto 1 - \frac{\color{blue}{1 \cdot x}}{\frac{8.333333333333334}{x}} \]

    2. div-inv 96.9%

       \[\leadsto 1 - \frac{1 \cdot x}{\color{blue}{8.333333333333334 \cdot \frac{1}{x}}} \]

    3. times-frac 97.0%

       \[\leadsto 1 - \color{blue}{\frac{1}{8.333333333333334} \cdot \frac{x}{\frac{1}{x}}} \]

    4. metadata-eval 97.0%

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

11. Applied egg-rr 97.0%

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

12. Final simplification 97.0%

    \[\leadsto 1 - 0.12 \cdot \frac{x}{\frac{1}{x}} \]
13. 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 99.8%

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

3. Final simplification 99.8%

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

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
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. Taylor expanded in x around inf 96.9%

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

   1. *-commutative 96.9%

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

6. Simplified 96.9%

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

7. Final simplification 96.9%

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

Alternative 6: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{x}{\frac{8.333333333333334}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (/ x (/ 8.333333333333334 x))))

C

double code(double x) {
	return 1.0 - (x / (8.333333333333334 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x / (8.333333333333334d0 / x))
end function

Java

public static double code(double x) {
	return 1.0 - (x / (8.333333333333334 / x));
}

Python

def code(x):
	return 1.0 - (x / (8.333333333333334 / x))

Julia

function code(x)
	return Float64(1.0 - Float64(x / Float64(8.333333333333334 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x / (8.333333333333334 / x));
end

Wolfram

code[x_] := N[(1.0 - N[(x / N[(8.333333333333334 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
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/ 93.0%

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

   3. metadata-eval 93.0%

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

   4. swap-sqr 93.0%

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

   5. pow2 93.0%

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

   6. metadata-eval 93.0%

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

   7. *-commutative 93.0%

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

   8. cancel-sign-sub-inv 93.0%

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

   9. metadata-eval 93.0%

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

4. Applied egg-rr 93.0%

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

   1. clear-num 99.8%

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

   2. un-div-inv 99.8%

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

   3. clear-num 99.8%

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

   4. metadata-eval 99.8%

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

   5. *-commutative 99.8%

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

   6. metadata-eval 99.8%

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

   7. unpow2 99.8%

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

   8. swap-sqr 99.8%

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

   9. *-commutative 99.8%

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

   10. flip-- 99.8%

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

   11. cancel-sign-sub-inv 99.8%

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

   12. metadata-eval 99.8%

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

   13. *-commutative 99.8%

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

   14. +-commutative 99.8%

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

   15. fma-define 99.9%

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

8. Applied egg-rr 99.9%

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

9. Taylor expanded in x around inf 96.9%

   \[\leadsto 1 - \frac{x}{\color{blue}{\frac{8.333333333333334}{x}}} \]

10. Final simplification 96.9%

    \[\leadsto 1 - \frac{x}{\frac{8.333333333333334}{x}} \]
11. Add Preprocessing

Alternative 7: 51.9% 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 99.8%

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

3. Taylor expanded in x around 0 52.8%

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

   1. *-commutative 52.8%

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

5. Simplified 52.8%

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

6. Final simplification 52.8%

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

Alternative 8: 49.9% 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 99.8%

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

3. Taylor expanded in x around 0 52.8%

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

   1. *-commutative 52.8%

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

5. Simplified 52.8%

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

6. Taylor expanded in x around 0 50.8%

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

7. Final simplification 50.8%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024078 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))