Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B

Percentage Accurate: 99.9% → 99.9%

Time: 12.9s

Alternatives: 11

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))

C

double code(double x) {
	return 0.70711 * (((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 = 0.70711d0 * (((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Python

def code(x):
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x)

Julia

function code(x)
	return Float64(0.70711 * 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 = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
end

Wolfram

code[x_] := N[(0.70711 * 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]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))

C

double code(double x) {
	return 0.70711 * (((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 = 0.70711d0 * (((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Python

def code(x):
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x)

Julia

function code(x)
	return Float64(0.70711 * 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 = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
end

Wolfram

code[x_] := N[(0.70711 * 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]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(x, 0.27061, 2.30753\right), \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot 0.70711, x \cdot -0.70711\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (fma x 0.27061 2.30753)
  (* (/ 1.0 (fma x (fma x 0.04481 0.99229) 1.0)) 0.70711)
  (* x -0.70711)))

C

double code(double x) {
	return fma(fma(x, 0.27061, 2.30753), ((1.0 / fma(x, fma(x, 0.04481, 0.99229), 1.0)) * 0.70711), (x * -0.70711));
}

Julia

function code(x)
	return fma(fma(x, 0.27061, 2.30753), Float64(Float64(1.0 / fma(x, fma(x, 0.04481, 0.99229), 1.0)) * 0.70711), Float64(x * -0.70711))
end

Wolfram

code[x_] := N[(N[(x * 0.27061 + 2.30753), $MachinePrecision] * N[(N[(1.0 / N[(x * N[(x * 0.04481 + 0.99229), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.70711), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

   3. clear-num N/A

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\frac{1}{\frac{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{70711}{100000}}}{\frac{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}} \cdot \frac{70711}{100000}} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

   7. clear-num N/A

      \[\leadsto \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

   8. div-inv N/A

      \[\leadsto \color{blue}{\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right) \cdot \frac{1}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}\right)} \cdot \frac{70711}{100000} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right) \cdot \left(\frac{1}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

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

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}, \frac{1}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}, \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.27061, 2.30753\right), \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot 0.70711, -0.70711 \cdot x\right)} \]

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, 0.27061, 2.30753\right), \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot 0.70711, x \cdot -0.70711\right) \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (fma x 0.27061 2.30753) (fma x (fma x 0.04481 0.99229) 1.0)) x)))

C

double code(double x) {
	return 0.70711 * ((fma(x, 0.27061, 2.30753) / fma(x, fma(x, 0.04481, 0.99229), 1.0)) - x);
}

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(fma(x, 0.27061, 2.30753) / fma(x, fma(x, 0.04481, 0.99229), 1.0)) - x))
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(x * 0.27061 + 2.30753), $MachinePrecision] / N[(x * N[(x * 0.04481 + 0.99229), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \cdot \frac{70711}{100000}} \]

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

      \[\leadsto \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \cdot \frac{70711}{100000}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \cdot \frac{70711}{100000} \]

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

      \[\leadsto \left(\color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x\right) \cdot \frac{70711}{100000} \]

   5. +-commutative N/A

      \[\leadsto \left(\frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \cdot \frac{70711}{100000} \]

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

      \[\leadsto \left(\frac{\color{blue}{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \cdot \frac{70711}{100000} \]

   7. +-commutative N/A

      \[\leadsto \left(\frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x\right) \cdot \frac{70711}{100000} \]

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

      \[\leadsto \left(\frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\color{blue}{\mathsf{fma}\left(x, \frac{99229}{100000} + x \cdot \frac{4481}{100000}, 1\right)}} - x\right) \cdot \frac{70711}{100000} \]

   9. +-commutative N/A

      \[\leadsto \left(\frac{\mathsf{fma}\left(x, \frac{27061}{100000}, \frac{230753}{100000}\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, 1\right)} - x\right) \cdot \frac{70711}{100000} \]

   10. accelerator-lowering-fma.f64 99.8

       \[\leadsto \left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.04481, 0.99229\right)}, 1\right)} - x\right) \cdot 0.70711 \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} - x\right) \cdot 0.70711} \]

5. Final simplification 99.8%

   \[\leadsto 0.70711 \cdot \left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} - x\right) \]
6. Add Preprocessing

Alternative 3: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.2692862305735844, 1.3436228731669864\right), -2.134856267379707\right), 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-0.70711 + \frac{4.2702753202410175}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (fma x -0.70711 (/ 4.2702753202410175 x))
   (if (<= x 0.85)
     (fma
      x
      (fma x (fma x -1.2692862305735844 1.3436228731669864) -2.134856267379707)
      1.6316775383)
     (* x (+ -0.70711 (/ 4.2702753202410175 (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = fma(x, -0.70711, (4.2702753202410175 / x));
	} else if (x <= 0.85) {
		tmp = fma(x, fma(x, fma(x, -1.2692862305735844, 1.3436228731669864), -2.134856267379707), 1.6316775383);
	} else {
		tmp = x * (-0.70711 + (4.2702753202410175 / (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = fma(x, -0.70711, Float64(4.2702753202410175 / x));
	elseif (x <= 0.85)
		tmp = fma(x, fma(x, fma(x, -1.2692862305735844, 1.3436228731669864), -2.134856267379707), 1.6316775383);
	else
		tmp = Float64(x * Float64(-0.70711 + Float64(4.2702753202410175 / Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.85], N[(x * N[(x * N[(x * -1.2692862305735844 + 1.3436228731669864), $MachinePrecision] + -2.134856267379707), $MachinePrecision] + 1.6316775383), $MachinePrecision], N[(x * N[(-0.70711 + N[(4.2702753202410175 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 99.7%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{-70711}{100000}}\right) \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-70711}{100000} + \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x + \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{-70711}{100000}} + \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x \]

      6. remove-double-neg N/A

         \[\leadsto x \cdot \frac{-70711}{100000} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right)\right)} \]

      7. distribute-lft-neg-out N/A

         \[\leadsto x \cdot \frac{-70711}{100000} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \frac{-70711}{100000} + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto x \cdot \frac{-70711}{100000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \frac{-70711}{100000} + \color{blue}{\left(-1 \cdot x\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-70711}{100000}, \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\left(x \cdot -1\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{x \cdot \left(-1 \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \frac{1913510371}{448100000}\right)}\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\left(x \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{1913510371}{448100000}}\right) \]

   5. Simplified 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)} \]

   if -1.05000000000000004 < x < 0.849999999999999978

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) + \frac{16316775383}{10000000000}} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}, \frac{16316775383}{10000000000}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}, \frac{16316775383}{10000000000}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \color{blue}{\frac{-2134856267379707}{1000000000000000}}, \frac{16316775383}{10000000000}\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x, \frac{-2134856267379707}{1000000000000000}\right)}, \frac{16316775383}{10000000000}\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x + \frac{134362287316698645903}{100000000000000000000}}, \frac{-2134856267379707}{1000000000000000}\right), \frac{16316775383}{10000000000}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-12692862305735843227608787}{10000000000000000000000000}} + \frac{134362287316698645903}{100000000000000000000}, \frac{-2134856267379707}{1000000000000000}\right), \frac{16316775383}{10000000000}\right) \]

      8. accelerator-lowering-fma.f64 99.3

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -1.2692862305735844, 1.3436228731669864\right)}, -2.134856267379707\right), 1.6316775383\right) \]

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.2692862305735844, 1.3436228731669864\right), -2.134856267379707\right), 1.6316775383\right)} \]

   if 0.849999999999999978 < x

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\mathsf{neg}\left(x\right)\right)\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{\frac{70711}{100000} \cdot \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

      3. clear-num N/A

         \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\frac{1}{\frac{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{70711}{100000}}{\frac{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \frac{70711}{100000}}}{\frac{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}} \cdot \frac{70711}{100000}} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      7. clear-num N/A

         \[\leadsto \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} \cdot \frac{70711}{100000} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      8. div-inv N/A

         \[\leadsto \color{blue}{\left(\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right) \cdot \frac{1}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}\right)} \cdot \frac{70711}{100000} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}\right) \cdot \left(\frac{1}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}\right)} + \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{230753}{100000} + x \cdot \frac{27061}{100000}, \frac{1}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} \cdot \frac{70711}{100000}, \frac{70711}{100000} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.27061, 2.30753\right), \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot 0.70711, -0.70711 \cdot x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{-70711}{100000}}\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-70711}{100000} + \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-70711}{100000} + \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto x \cdot \left(\frac{-70711}{100000} + \color{blue}{\frac{\frac{1913510371}{448100000} \cdot 1}{{x}^{2}}}\right) \]

      7. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{-70711}{100000} + \frac{\color{blue}{\frac{1913510371}{448100000}}}{{x}^{2}}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \left(\frac{-70711}{100000} + \color{blue}{\frac{\frac{1913510371}{448100000}}{{x}^{2}}}\right) \]

      9. unpow2 N/A

         \[\leadsto x \cdot \left(\frac{-70711}{100000} + \frac{\frac{1913510371}{448100000}}{\color{blue}{x \cdot x}}\right) \]

      10. *-lowering-*.f64 97.8

          \[\leadsto x \cdot \left(-0.70711 + \frac{4.2702753202410175}{\color{blue}{x \cdot x}}\right) \]

   7. Simplified 97.8%

      \[\leadsto \color{blue}{x \cdot \left(-0.70711 + \frac{4.2702753202410175}{x \cdot x}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.85:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.2692862305735844, 1.3436228731669864\right), -2.134856267379707\right), 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x -0.70711 (/ 4.2702753202410175 x))))
   (if (<= x -1.05)
     t_0
     (if (<= x 0.85)
       (fma
        x
        (fma
         x
         (fma x -1.2692862305735844 1.3436228731669864)
         -2.134856267379707)
        1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = fma(x, -0.70711, (4.2702753202410175 / x));
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 0.85) {
		tmp = fma(x, fma(x, fma(x, -1.2692862305735844, 1.3436228731669864), -2.134856267379707), 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, -0.70711, Float64(4.2702753202410175 / x))
	tmp = 0.0
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 0.85)
		tmp = fma(x, fma(x, fma(x, -1.2692862305735844, 1.3436228731669864), -2.134856267379707), 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05], t$95$0, If[LessEqual[x, 0.85], N[(x * N[(x * N[(x * -1.2692862305735844 + 1.3436228731669864), $MachinePrecision] + -2.134856267379707), $MachinePrecision] + 1.6316775383), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 0.849999999999999978 < x

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{-70711}{100000}}\right) \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-70711}{100000} + \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x + \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{-70711}{100000}} + \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x \]

      6. remove-double-neg N/A

         \[\leadsto x \cdot \frac{-70711}{100000} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right)\right)} \]

      7. distribute-lft-neg-out N/A

         \[\leadsto x \cdot \frac{-70711}{100000} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \frac{-70711}{100000} + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto x \cdot \frac{-70711}{100000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \frac{-70711}{100000} + \color{blue}{\left(-1 \cdot x\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-70711}{100000}, \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\left(x \cdot -1\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{x \cdot \left(-1 \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \frac{1913510371}{448100000}\right)}\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\left(x \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{1913510371}{448100000}}\right) \]

   5. Simplified 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)} \]

   if -1.05000000000000004 < x < 0.849999999999999978

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right) + \frac{16316775383}{10000000000}} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}, \frac{16316775383}{10000000000}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}, \frac{16316775383}{10000000000}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) + \color{blue}{\frac{-2134856267379707}{1000000000000000}}, \frac{16316775383}{10000000000}\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x, \frac{-2134856267379707}{1000000000000000}\right)}, \frac{16316775383}{10000000000}\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x + \frac{134362287316698645903}{100000000000000000000}}, \frac{-2134856267379707}{1000000000000000}\right), \frac{16316775383}{10000000000}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-12692862305735843227608787}{10000000000000000000000000}} + \frac{134362287316698645903}{100000000000000000000}, \frac{-2134856267379707}{1000000000000000}\right), \frac{16316775383}{10000000000}\right) \]

      8. accelerator-lowering-fma.f64 99.3

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -1.2692862305735844, 1.3436228731669864\right)}, -2.134856267379707\right), 1.6316775383\right) \]

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.2692862305735844, 1.3436228731669864\right), -2.134856267379707\right), 1.6316775383\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 1.3436228731669864, -2.134856267379707\right), 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x -0.70711 (/ 4.2702753202410175 x))))
   (if (<= x -1.05)
     t_0
     (if (<= x 1.6)
       (fma x (fma x 1.3436228731669864 -2.134856267379707) 1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = fma(x, -0.70711, (4.2702753202410175 / x));
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 1.6) {
		tmp = fma(x, fma(x, 1.3436228731669864, -2.134856267379707), 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, -0.70711, Float64(4.2702753202410175 / x))
	tmp = 0.0
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 1.6)
		tmp = fma(x, fma(x, 1.3436228731669864, -2.134856267379707), 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * -0.70711 + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05], t$95$0, If[LessEqual[x, 1.6], N[(x * N[(x * 1.3436228731669864 + -2.134856267379707), $MachinePrecision] + 1.6316775383), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.6000000000000001 < x

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{-70711}{100000}}\right) \]

      3. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-70711}{100000} + \frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x + \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{-70711}{100000}} + \left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x \]

      6. remove-double-neg N/A

         \[\leadsto x \cdot \frac{-70711}{100000} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right)\right)} \]

      7. distribute-lft-neg-out N/A

         \[\leadsto x \cdot \frac{-70711}{100000} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \frac{-70711}{100000} + \left(\mathsf{neg}\left(\color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto x \cdot \frac{-70711}{100000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot \frac{-70711}{100000} + \color{blue}{\left(-1 \cdot x\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right) \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-70711}{100000}, \left(-1 \cdot x\right) \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\left(x \cdot -1\right)} \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{x \cdot \left(-1 \cdot \left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, x \cdot \color{blue}{\left(\frac{1913510371}{448100000} \cdot \frac{1}{{x}^{2}}\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \frac{1913510371}{448100000}\right)}\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\left(x \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{1913510371}{448100000}}\right) \]

   5. Simplified 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{4.2702753202410175}{x}\right)} \]

   if -1.05000000000000004 < x < 1.6000000000000001

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right) + \frac{16316775383}{10000000000}} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}, \frac{16316775383}{10000000000}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{134362287316698645903}{100000000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}, \frac{16316775383}{10000000000}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{134362287316698645903}{100000000000000000000}} + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), \frac{16316775383}{10000000000}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{134362287316698645903}{100000000000000000000} + \color{blue}{\frac{-2134856267379707}{1000000000000000}}, \frac{16316775383}{10000000000}\right) \]

      6. accelerator-lowering-fma.f64 99.2

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 1.3436228731669864, -2.134856267379707\right)}, 1.6316775383\right) \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 1.3436228731669864, -2.134856267379707\right), 1.6316775383\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, -0.70711, \frac{1.644355519354221}{x}\right)\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 1.3436228731669864, -2.134856267379707\right), 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x -0.70711 (/ 1.644355519354221 x))))
   (if (<= x -1.05)
     t_0
     (if (<= x 1.0)
       (fma x (fma x 1.3436228731669864 -2.134856267379707) 1.6316775383)
       t_0))))

C

double code(double x) {
	double t_0 = fma(x, -0.70711, (1.644355519354221 / x));
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = fma(x, fma(x, 1.3436228731669864, -2.134856267379707), 1.6316775383);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, -0.70711, Float64(1.644355519354221 / x))
	tmp = 0.0
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = fma(x, fma(x, 1.3436228731669864, -2.134856267379707), 1.6316775383);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * -0.70711 + N[(1.644355519354221 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05], t$95$0, If[LessEqual[x, 1.0], N[(x * N[(x * 1.3436228731669864 + -2.134856267379707), $MachinePrecision] + 1.6316775383), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1 < x

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \color{blue}{\frac{99229}{100000}}} - x\right) \]
   4. Step-by-step derivation

   5. Simplified 97.3%

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \color{blue}{0.99229}} - x\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\color{blue}{\frac{230753}{100000}}}{1 + x \cdot \frac{99229}{100000}} - x\right) \]
   7. Step-by-step derivation

   8. Simplified 97.9%

      \[\leadsto 0.70711 \cdot \left(\frac{\color{blue}{2.30753}}{1 + x \cdot 0.99229} - x\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\frac{\frac{230753}{99229} - \frac{23075300000}{9846394441} \cdot \frac{1}{x}}{x}} - x\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{70711}{100000} \cdot \left(\color{blue}{\frac{\frac{230753}{99229} - \frac{23075300000}{9846394441} \cdot \frac{1}{x}}{x}} - x\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\color{blue}{\frac{230753}{99229} - \frac{23075300000}{9846394441} \cdot \frac{1}{x}}}{x} - x\right) \]

       3. associate-*r/ N/A

          \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{99229} - \color{blue}{\frac{\frac{23075300000}{9846394441} \cdot 1}{x}}}{x} - x\right) \]

       4. metadata-eval N/A

          \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{99229} - \frac{\color{blue}{\frac{23075300000}{9846394441}}}{x}}{x} - x\right) \]

       5. /-lowering-/.f64 98.0

          \[\leadsto 0.70711 \cdot \left(\frac{2.3254592911346483 - \color{blue}{\frac{2.343527891175612}{x}}}{x} - x\right) \]

   11. Simplified 98.0%

       \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{2.3254592911346483 - \frac{2.343527891175612}{x}}{x}} - x\right) \]

   12. Taylor expanded in x around inf

       \[\leadsto \color{blue}{x \cdot \left(\frac{16316775383}{9922900000} \cdot \frac{1}{{x}^{2}} - \frac{70711}{100000}\right)} \]
   13. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{16316775383}{9922900000} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{70711}{100000}\right)\right)\right)} \]

       2. metadata-eval N/A

          \[\leadsto x \cdot \left(\frac{16316775383}{9922900000} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{-70711}{100000}}\right) \]

       3. +-commutative N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{-70711}{100000} + \frac{16316775383}{9922900000} \cdot \frac{1}{{x}^{2}}\right)} \]

       4. distribute-rgt-in N/A

          \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x + \left(\frac{16316775383}{9922900000} \cdot \frac{1}{{x}^{2}}\right) \cdot x} \]

       5. *-commutative N/A

          \[\leadsto \color{blue}{x \cdot \frac{-70711}{100000}} + \left(\frac{16316775383}{9922900000} \cdot \frac{1}{{x}^{2}}\right) \cdot x \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-70711}{100000}, \left(\frac{16316775383}{9922900000} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)} \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\frac{\frac{16316775383}{9922900000} \cdot 1}{{x}^{2}}} \cdot x\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{\frac{16316775383}{9922900000}}}{{x}^{2}} \cdot x\right) \]

       9. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\frac{\frac{16316775383}{9922900000} \cdot x}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{16316775383}{9922900000} \cdot x}{\color{blue}{x \cdot x}}\right) \]

       11. associate-/r* N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \color{blue}{\frac{\frac{\frac{16316775383}{9922900000} \cdot x}{x}}{x}}\right) \]

       12. associate-/l* N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{\frac{16316775383}{9922900000} \cdot \frac{x}{x}}}{x}\right) \]

       13. *-inverses N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\frac{16316775383}{9922900000} \cdot \color{blue}{1}}{x}\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{-70711}{100000}, \frac{\color{blue}{\frac{16316775383}{9922900000}}}{x}\right) \]

       15. /-lowering-/.f64 97.9

           \[\leadsto \mathsf{fma}\left(x, -0.70711, \color{blue}{\frac{1.644355519354221}{x}}\right) \]

   14. Simplified 97.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{1.644355519354221}{x}\right)} \]

   if -1.05000000000000004 < x < 1

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right) + \frac{16316775383}{10000000000}} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}, \frac{16316775383}{10000000000}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{134362287316698645903}{100000000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}, \frac{16316775383}{10000000000}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{134362287316698645903}{100000000000000000000}} + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), \frac{16316775383}{10000000000}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{134362287316698645903}{100000000000000000000} + \color{blue}{\frac{-2134856267379707}{1000000000000000}}, \frac{16316775383}{10000000000}\right) \]

      6. accelerator-lowering-fma.f64 99.2

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 1.3436228731669864, -2.134856267379707\right)}, 1.6316775383\right) \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 1.3436228731669864, -2.134856267379707\right), 1.6316775383\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{2.30753}{1 + x \cdot 0.99229} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.70711 (- (/ 2.30753 (+ 1.0 (* x 0.99229))) x)))

C

double code(double x) {
	return 0.70711 * ((2.30753 / (1.0 + (x * 0.99229))) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.70711d0 * ((2.30753d0 / (1.0d0 + (x * 0.99229d0))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * ((2.30753 / (1.0 + (x * 0.99229))) - x);
}

Python

def code(x):
	return 0.70711 * ((2.30753 / (1.0 + (x * 0.99229))) - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(2.30753 / Float64(1.0 + Float64(x * 0.99229))) - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * ((2.30753 / (1.0 + (x * 0.99229))) - x);
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(2.30753 / N[(1.0 + N[(x * 0.99229), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \color{blue}{\frac{99229}{100000}}} - x\right) \]
4. Step-by-step derivation

5. Simplified 98.0%

   \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \color{blue}{0.99229}} - x\right) \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{70711}{100000} \cdot \left(\frac{\color{blue}{\frac{230753}{100000}}}{1 + x \cdot \frac{99229}{100000}} - x\right) \]
7. Step-by-step derivation

8. Simplified 98.1%

   \[\leadsto 0.70711 \cdot \left(\frac{\color{blue}{2.30753}}{1 + x \cdot 0.99229} - x\right) \]
9. Add Preprocessing

Alternative 8: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.16:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 1.3436228731669864, -2.134856267379707\right), 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* x -0.70711)
   (if (<= x 1.16)
     (fma x (fma x 1.3436228731669864 -2.134856267379707) 1.6316775383)
     (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.16) {
		tmp = fma(x, fma(x, 1.3436228731669864, -2.134856267379707), 1.6316775383);
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(x * -0.70711);
	elseif (x <= 1.16)
		tmp = fma(x, fma(x, 1.3436228731669864, -2.134856267379707), 1.6316775383);
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 1.16], N[(x * N[(x * 1.3436228731669864 + -2.134856267379707), $MachinePrecision] + 1.6316775383), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.15999999999999992 < x

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{-70711}{100000}} \]

      2. *-lowering-*.f64 97.9

         \[\leadsto \color{blue}{x \cdot -0.70711} \]

   5. Simplified 97.9%

      \[\leadsto \color{blue}{x \cdot -0.70711} \]

   if -1.05000000000000004 < x < 1.15999999999999992

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}\right) + \frac{16316775383}{10000000000}} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{134362287316698645903}{100000000000000000000} \cdot x - \frac{2134856267379707}{1000000000000000}, \frac{16316775383}{10000000000}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{134362287316698645903}{100000000000000000000} \cdot x + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right)}, \frac{16316775383}{10000000000}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{134362287316698645903}{100000000000000000000}} + \left(\mathsf{neg}\left(\frac{2134856267379707}{1000000000000000}\right)\right), \frac{16316775383}{10000000000}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{134362287316698645903}{100000000000000000000} + \color{blue}{\frac{-2134856267379707}{1000000000000000}}, \frac{16316775383}{10000000000}\right) \]

      6. accelerator-lowering-fma.f64 99.2

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 1.3436228731669864, -2.134856267379707\right)}, 1.6316775383\right) \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 1.3436228731669864, -2.134856267379707\right), 1.6316775383\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 98.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;\mathsf{fma}\left(x, -2.134856267379707, 1.6316775383\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* x -0.70711)
   (if (<= x 1.15) (fma x -2.134856267379707 1.6316775383) (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = fma(x, -2.134856267379707, 1.6316775383);
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(x * -0.70711);
	elseif (x <= 1.15)
		tmp = fma(x, -2.134856267379707, 1.6316775383);
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 1.15], N[(x * -2.134856267379707 + 1.6316775383), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < x

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{-70711}{100000}} \]

      2. *-lowering-*.f64 97.9

         \[\leadsto \color{blue}{x \cdot -0.70711} \]

   5. Simplified 97.9%

      \[\leadsto \color{blue}{x \cdot -0.70711} \]

   if -1.05000000000000004 < x < 1.1499999999999999

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + \frac{-2134856267379707}{1000000000000000} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-2134856267379707}{1000000000000000} \cdot x + \frac{16316775383}{10000000000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{-2134856267379707}{1000000000000000}} + \frac{16316775383}{10000000000} \]

      3. accelerator-lowering-fma.f64 99.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -2.134856267379707, 1.6316775383\right)} \]

   5. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -2.134856267379707, 1.6316775383\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 98.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.16:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.4) (* x -0.70711) (if (<= x 1.16) 1.6316775383 (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -3.4) {
		tmp = x * -0.70711;
	} else if (x <= 1.16) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.4d0)) then
        tmp = x * (-0.70711d0)
    else if (x <= 1.16d0) then
        tmp = 1.6316775383d0
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.4) {
		tmp = x * -0.70711;
	} else if (x <= 1.16) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.4:
		tmp = x * -0.70711
	elif x <= 1.16:
		tmp = 1.6316775383
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.4)
		tmp = Float64(x * -0.70711);
	elseif (x <= 1.16)
		tmp = 1.6316775383;
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.4)
		tmp = x * -0.70711;
	elseif (x <= 1.16)
		tmp = 1.6316775383;
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.4], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 1.16], 1.6316775383, N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.39999999999999991 or 1.15999999999999992 < x

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-70711}{100000} \cdot x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{-70711}{100000}} \]

      2. *-lowering-*.f64 98.4

         \[\leadsto \color{blue}{x \cdot -0.70711} \]

   5. Simplified 98.4%

      \[\leadsto \color{blue}{x \cdot -0.70711} \]

   if -3.39999999999999991 < x < 1.15999999999999992

   1. Initial program 99.9%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000}} \]
   4. Step-by-step derivation

   5. Simplified 97.4%

      \[\leadsto \color{blue}{1.6316775383} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 51.2% accurate, 44.0× speedup

Math

\[\begin{array}{l} \\ 1.6316775383 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.6316775383)

C

double code(double x) {
	return 1.6316775383;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.6316775383d0
end function

Java

public static double code(double x) {
	return 1.6316775383;
}

Python

def code(x):
	return 1.6316775383

Julia

function code(x)
	return 1.6316775383
end

MATLAB

function tmp = code(x)
	tmp = 1.6316775383;
end

Wolfram

code[x_] := 1.6316775383

Derivation

1. Initial program 99.8%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{16316775383}{10000000000}} \]
4. Step-by-step derivation

5. Simplified 43.4%

   \[\leadsto \color{blue}{1.6316775383} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))