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

Percentage Accurate: 100.0% → 100.0%

Time: 8.1s

Alternatives: 9

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.27061 \cdot x + 2.30753}{1 + \mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ (* 0.27061 x) 2.30753) (+ 1.0 (* (fma 0.04481 x 0.99229) x))) x))

C

double code(double x) {
	return (((0.27061 * x) + 2.30753) / (1.0 + (fma(0.04481, x, 0.99229) * x))) - x;
}

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.27061 * x) + 2.30753) / Float64(1.0 + Float64(fma(0.04481, x, 0.99229) * x))) - x)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(1.0 + N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

   3. lower-+.f64 100.0

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}} - x \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

   6. lower-*.f64 100.0

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\left(0.99229 + x \cdot 0.04481\right) \cdot x} + 1} - x \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot x + 1} - x \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(x \cdot \frac{4481}{100000} + \frac{99229}{100000}\right)} \cdot x + 1} - x \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}\right) \cdot x + 1} - x \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}\right) \cdot x + 1} - x \]

   11. lower-fma.f64 100.0

       \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)} \cdot x + 1} - x \]

4. Applied rewrites 100.0%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x + 1}} - x \]

5. Final simplification 100.0%

   \[\leadsto \frac{0.27061 \cdot x + 2.30753}{1 + \mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x} - x \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{1 + \mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (fma x 0.27061 2.30753) (+ 1.0 (* (fma 0.04481 x 0.99229) x))) x))

C

double code(double x) {
	return (fma(x, 0.27061, 2.30753) / (1.0 + (fma(0.04481, x, 0.99229) * x))) - x;
}

Julia

function code(x)
	return Float64(Float64(fma(x, 0.27061, 2.30753) / Float64(1.0 + Float64(fma(0.04481, x, 0.99229) * x))) - x)
end

Wolfram

code[x_] := N[(N[(N[(x * 0.27061 + 2.30753), $MachinePrecision] / N[(1.0 + N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

   3. lower-+.f64 100.0

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}} - x \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

   6. lower-*.f64 100.0

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\left(0.99229 + x \cdot 0.04481\right) \cdot x} + 1} - x \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot x + 1} - x \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(x \cdot \frac{4481}{100000} + \frac{99229}{100000}\right)} \cdot x + 1} - x \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}\right) \cdot x + 1} - x \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}\right) \cdot x + 1} - x \]

   11. lower-fma.f64 100.0

       \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)} \cdot x + 1} - x \]

4. Applied rewrites 100.0%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x + 1}} - x \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right) \cdot x + 1} - x \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right) \cdot x + 1} - x \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right) \cdot x + 1} - x \]

   4. lower-fma.f64 100.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x + 1} - x \]

6. Applied rewrites 100.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x + 1} - x \]

7. Final simplification 100.0%

   \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{1 + \mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x} - x \]
8. Add Preprocessing

Alternative 3: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.27061 \cdot x + 2.30753}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ (* 0.27061 x) 2.30753) (fma (fma 0.04481 x 0.99229) x 1.0)) x))

C

double code(double x) {
	return (((0.27061 * x) + 2.30753) / fma(fma(0.04481, x, 0.99229), x, 1.0)) - x;
}

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.27061 * x) + 2.30753) / fma(fma(0.04481, x, 0.99229), x, 1.0)) - x)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.27061 * x), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

   5. lower-fma.f64 100.0

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\mathsf{fma}\left(0.99229 + x \cdot 0.04481, x, 1\right)}} - x \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} + x \cdot \frac{4481}{100000}}, x, 1\right)} - x \]

   7. +-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, x, 1\right)} - x \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)} - x \]

   9. *-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}, x, 1\right)} - x \]

   10. lower-fma.f64 100.0

       \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)}, x, 1\right)} - x \]

4. Applied rewrites 100.0%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)}} - x \]

5. Final simplification 100.0%

   \[\leadsto \frac{0.27061 \cdot x + 2.30753}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x \]
6. Add Preprocessing

Alternative 4: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (fma 0.27061 x 2.30753) (fma (fma 0.04481 x 0.99229) x 1.0)) x))

C

double code(double x) {
	return (fma(0.27061, x, 2.30753) / fma(fma(0.04481, x, 0.99229), x, 1.0)) - x;
}

Julia

function code(x)
	return Float64(Float64(fma(0.27061, x, 2.30753) / fma(fma(0.04481, x, 0.99229), x, 1.0)) - x)
end

Wolfram

code[x_] := N[(N[(N[(0.27061 * x + 2.30753), $MachinePrecision] / N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   4. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   5. lower-fma.f64 100.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

   9. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

   10. lower-fma.f64 100.0

       \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(0.99229 + x \cdot 0.04481, x, 1\right)}} - x \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} + x \cdot \frac{4481}{100000}}, x, 1\right)} - x \]

   12. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, x, 1\right)} - x \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)} - x \]

   14. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}, x, 1\right)} - x \]

   15. lower-fma.f64 100.0

       \[\leadsto \frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)}, x, 1\right)} - x \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.27061, x, 2.30753\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)} - x} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.025050834237766436, x, 0.37920088514346545\right), x, 0.4333638132548656\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  (/
   1.0
   (fma
    (fma -0.025050834237766436 x 0.37920088514346545)
    x
    0.4333638132548656))
  x))

C

double code(double x) {
	return (1.0 / fma(fma(-0.025050834237766436, x, 0.37920088514346545), x, 0.4333638132548656)) - x;
}

Julia

function code(x)
	return Float64(Float64(1.0 / fma(fma(-0.025050834237766436, x, 0.37920088514346545), x, 0.4333638132548656)) - x)
end

Wolfram

code[x_] := N[(N[(1.0 / N[(N[(-0.025050834237766436 * x + 0.37920088514346545), $MachinePrecision] * x + 0.4333638132548656), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

   3. lower-+.f64 100.0

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}} - x \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

   6. lower-*.f64 100.0

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\left(0.99229 + x \cdot 0.04481\right) \cdot x} + 1} - x \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot x + 1} - x \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(x \cdot \frac{4481}{100000} + \frac{99229}{100000}\right)} \cdot x + 1} - x \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}\right) \cdot x + 1} - x \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}\right) \cdot x + 1} - x \]

   11. lower-fma.f64 100.0

       \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)} \cdot x + 1} - x \]

4. Applied rewrites 100.0%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x + 1}} - x \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right) \cdot x + 1} - x \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right) \cdot x + 1} - x \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right) \cdot x + 1} - x \]

   4. lower-fma.f64 100.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x + 1} - x \]

6. Applied rewrites 100.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x + 1} - x \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right) \cdot x + 1}} - x \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right) \cdot x + 1}{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}}} - x \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right) \cdot x + 1}{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}}} - x \]

   4. lower-/.f64 100.0

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x + 1}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}} - x \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right) \cdot x + 1}}{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}} - x \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right) \cdot x} + 1}{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}} - x \]

   7. lower-fma.f64 100.0

      \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)}}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}} - x \]

   8. lift-fma.f64 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x + \frac{99229}{100000}}, x, 1\right)}{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}} - x \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)}{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}} - x \]

   10. lower-fma.f64 100.0

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, 0.04481, 0.99229\right)}, x, 1\right)}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}} - x \]

   11. lift-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), x, 1\right)}{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}} - x \]

   12. *-commutative N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{4481}{100000}, \frac{99229}{100000}\right), x, 1\right)}{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}} - x \]

   13. lower-fma.f64 100.0

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1\right)}{\color{blue}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}} - x \]

8. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.04481, 0.99229\right), x, 1\right)}{\mathsf{fma}\left(0.27061, x, 2.30753\right)}}} - x \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{100000}{230753} + x \cdot \left(\frac{20191289437}{53246947009} + \frac{-307796913907328}{12286892763167777} \cdot x\right)}} - x \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\frac{20191289437}{53246947009} + \frac{-307796913907328}{12286892763167777} \cdot x\right) + \frac{100000}{230753}}} - x \]

    2. *-commutative N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\frac{20191289437}{53246947009} + \frac{-307796913907328}{12286892763167777} \cdot x\right) \cdot x} + \frac{100000}{230753}} - x \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{20191289437}{53246947009} + \frac{-307796913907328}{12286892763167777} \cdot x, x, \frac{100000}{230753}\right)}} - x \]

    4. +-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{-307796913907328}{12286892763167777} \cdot x + \frac{20191289437}{53246947009}}, x, \frac{100000}{230753}\right)} - x \]

    5. lower-fma.f64 99.0

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.025050834237766436, x, 0.37920088514346545\right)}, x, 0.4333638132548656\right)} - x \]

11. Applied rewrites 99.0%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.025050834237766436, x, 0.37920088514346545\right), x, 0.4333638132548656\right)}} - x \]
12. Add Preprocessing

Alternative 6: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{2.30753}{\mathsf{fma}\left(0.99229, x, 1\right)} - x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 2.30753 (fma 0.99229 x 1.0)) x))

C

double code(double x) {
	return (2.30753 / fma(0.99229, x, 1.0)) - x;
}

Julia

function code(x)
	return Float64(Float64(2.30753 / fma(0.99229, x, 1.0)) - x)
end

Wolfram

code[x_] := N[(N[(2.30753 / N[(0.99229 * x + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

   3. lower-+.f64 100.0

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}} - x \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

   6. lower-*.f64 100.0

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\left(0.99229 + x \cdot 0.04481\right) \cdot x} + 1} - x \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot x + 1} - x \]

   8. +-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(x \cdot \frac{4481}{100000} + \frac{99229}{100000}\right)} \cdot x + 1} - x \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}\right) \cdot x + 1} - x \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}\right) \cdot x + 1} - x \]

   11. lower-fma.f64 100.0

       \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right)} \cdot x + 1} - x \]

4. Applied rewrites 100.0%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x + 1}} - x \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right) \cdot x + 1} - x \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right) \cdot x + 1} - x \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right) \cdot x + 1} - x \]

   4. lower-fma.f64 100.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x + 1} - x \]

6. Applied rewrites 100.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{\mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x + 1} - x \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{230753}{100000}}}{\mathsf{fma}\left(\frac{4481}{100000}, x, \frac{99229}{100000}\right) \cdot x + 1} - x \]
8. Step-by-step derivation

9. Applied rewrites 98.3%

   \[\leadsto \frac{\color{blue}{2.30753}}{\mathsf{fma}\left(0.04481, x, 0.99229\right) \cdot x + 1} - x \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{\frac{230753}{100000}}{\color{blue}{1 + \frac{99229}{100000} \cdot x}} - x \]
11. Step-by-step derivation

    1. *-lft-identity N/A

       \[\leadsto \frac{\frac{230753}{100000}}{1 + \frac{99229}{100000} \cdot \color{blue}{\left(1 \cdot x\right)}} - x \]

    2. lft-mult-inverse N/A

       \[\leadsto \frac{\frac{230753}{100000}}{1 + \frac{99229}{100000} \cdot \left(\color{blue}{\left(\frac{1}{x} \cdot x\right)} \cdot x\right)} - x \]

    3. associate-*r* N/A

       \[\leadsto \frac{\frac{230753}{100000}}{1 + \frac{99229}{100000} \cdot \color{blue}{\left(\frac{1}{x} \cdot \left(x \cdot x\right)\right)}} - x \]

    4. unpow2 N/A

       \[\leadsto \frac{\frac{230753}{100000}}{1 + \frac{99229}{100000} \cdot \left(\frac{1}{x} \cdot \color{blue}{{x}^{2}}\right)} - x \]

    5. associate-*l* N/A

       \[\leadsto \frac{\frac{230753}{100000}}{1 + \color{blue}{\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot {x}^{2}}} - x \]

    6. *-commutative N/A

       \[\leadsto \frac{\frac{230753}{100000}}{1 + \color{blue}{{x}^{2} \cdot \left(\frac{99229}{100000} \cdot \frac{1}{x}\right)}} - x \]

    7. +-commutative N/A

       \[\leadsto \frac{\frac{230753}{100000}}{\color{blue}{{x}^{2} \cdot \left(\frac{99229}{100000} \cdot \frac{1}{x}\right) + 1}} - x \]

    8. *-commutative N/A

       \[\leadsto \frac{\frac{230753}{100000}}{\color{blue}{\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot {x}^{2}} + 1} - x \]

    9. unpow2 N/A

       \[\leadsto \frac{\frac{230753}{100000}}{\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot \color{blue}{\left(x \cdot x\right)} + 1} - x \]

    10. associate-*r* N/A

        \[\leadsto \frac{\frac{230753}{100000}}{\color{blue}{\left(\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x\right) \cdot x} + 1} - x \]

    11. lower-fma.f64 N/A

        \[\leadsto \frac{\frac{230753}{100000}}{\color{blue}{\mathsf{fma}\left(\left(\frac{99229}{100000} \cdot \frac{1}{x}\right) \cdot x, x, 1\right)}} - x \]

    12. associate-*l* N/A

        \[\leadsto \frac{\frac{230753}{100000}}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} \cdot \left(\frac{1}{x} \cdot x\right)}, x, 1\right)} - x \]

    13. lft-mult-inverse N/A

        \[\leadsto \frac{\frac{230753}{100000}}{\mathsf{fma}\left(\frac{99229}{100000} \cdot \color{blue}{1}, x, 1\right)} - x \]

    14. metadata-eval 98.3

        \[\leadsto \frac{2.30753}{\mathsf{fma}\left(\color{blue}{0.99229}, x, 1\right)} - x \]

12. Applied rewrites 98.3%

    \[\leadsto \frac{2.30753}{\color{blue}{\mathsf{fma}\left(0.99229, x, 1\right)}} - x \]
13. Add Preprocessing

Alternative 7: 98.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05) (- x) (if (<= x 1.2) 2.30753 (- x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = -x;
	} else if (x <= 1.2) {
		tmp = 2.30753;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = -x
    else if (x <= 1.2d0) then
        tmp = 2.30753d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = -x;
	} else if (x <= 1.2) {
		tmp = 2.30753;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = -x
	elif x <= 1.2:
		tmp = 2.30753
	else:
		tmp = -x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(-x);
	elseif (x <= 1.2)
		tmp = 2.30753;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = -x;
	elseif (x <= 1.2)
		tmp = 2.30753;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], (-x), If[LessEqual[x, 1.2], 2.30753, (-x)]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.19999999999999996 < x

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 99.0

         \[\leadsto \color{blue}{-x} \]

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{-x} \]

   if -1.05000000000000004 < x < 1.19999999999999996

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{230753}{100000}} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.0%

      \[\leadsto \color{blue}{2.30753} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.0% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ 2.30753 - x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 2.30753 x))

C

double code(double x) {
	return 2.30753 - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.30753d0 - x
end function

Java

public static double code(double x) {
	return 2.30753 - x;
}

Python

def code(x):
	return 2.30753 - x

Julia

function code(x)
	return Float64(2.30753 - x)
end

MATLAB

function tmp = code(x)
	tmp = 2.30753 - x;
end

Wolfram

code[x_] := N[(2.30753 - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{230753}{100000}} - x \]
4. Step-by-step derivation

5. Applied rewrites 97.6%

   \[\leadsto \color{blue}{2.30753} - x \]
6. Add Preprocessing

Alternative 9: 50.0% accurate, 39.0× speedup

Math

\[\begin{array}{l} \\ 2.30753 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.30753)

C

double code(double x) {
	return 2.30753;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.30753d0
end function

Java

public static double code(double x) {
	return 2.30753;
}

Python

def code(x):
	return 2.30753

Julia

function code(x)
	return 2.30753
end

MATLAB

function tmp = code(x)
	tmp = 2.30753;
end

Wolfram

code[x_] := 2.30753

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{230753}{100000}} \]
4. Step-by-step derivation

5. Applied rewrites 49.2%

   \[\leadsto \color{blue}{2.30753} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024277 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  :precision binary64
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))