Logistic function from Lakshay Garg

Percentage Accurate: 54.0% → 99.9%

Time: 8.3s

Alternatives: 14

Speedup: 3.7×

Specification

Math

\[\begin{array}{l} \\ \frac{2}{1 + e^{-2 \cdot x}} - 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))

C

double code(double x, double y) {
	return (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) - 1.0d0
end function

Java

public static double code(double x, double y) {
	return (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
}

Python

def code(x, y):
	return (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0

Julia

function code(x, y)
	return Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
end

Wolfram

code[x_, y_] := N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 54.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{1 + e^{-2 \cdot x}} - 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))

C

double code(double x, double y) {
	return (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) - 1.0d0
end function

Java

public static double code(double x, double y) {
	return (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
}

Python

def code(x, y):
	return (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0

Julia

function code(x, y)
	return Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
end

Wolfram

code[x_, y_] := N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -2:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)\\ \mathbf{elif}\;x \cdot -2 \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) -2.0)
   (expm1 (- (log 2.0) (log1p (pow (exp x) -2.0))))
   (if (<= (* x -2.0) 0.0005)
     (fma
      (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x x))
      x
      x)
     (- (/ 2.0 (+ (exp (* x -2.0)) 1.0)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= -2.0) {
		tmp = expm1((log(2.0) - log1p(pow(exp(x), -2.0))));
	} else if ((x * -2.0) <= 0.0005) {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * (x * x)), x, x);
	} else {
		tmp = (2.0 / (exp((x * -2.0)) + 1.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= -2.0)
		tmp = expm1(Float64(log(2.0) - log1p((exp(x) ^ -2.0))));
	elseif (Float64(x * -2.0) <= 0.0005)
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * Float64(x * x)), x, x);
	else
		tmp = Float64(Float64(2.0 / Float64(exp(Float64(x * -2.0)) + 1.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], -2.0], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[N[(x * -2.0), $MachinePrecision], 0.0005], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(2.0 / N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -2

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}} - 1 \]

      7. lower-expm1.f64 N/A

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}\right) \]

      9. log-div N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(\log \left(1 + e^{-2 \cdot x}\right) - \log 2\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(\log \left(1 + e^{-2 \cdot x}\right) - \log 2\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\left(\log \color{blue}{\left(1 + e^{-2 \cdot x}\right)} - \log 2\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      12. lower-log1p.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\left(\color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)} - \log 2\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      13. lift-exp.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\color{blue}{e^{-2 \cdot x}}\right) - \log 2\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\left(\mathsf{log1p}\left(e^{\color{blue}{-2 \cdot x}}\right) - \log 2\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{expm1}\left(\left(\mathsf{log1p}\left(e^{\color{blue}{x \cdot -2}}\right) - \log 2\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      16. exp-prod N/A

          \[\leadsto \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\color{blue}{{\left(e^{x}\right)}^{-2}}\right) - \log 2\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      17. lower-pow.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\color{blue}{{\left(e^{x}\right)}^{-2}}\right) - \log 2\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      18. lower-exp.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\left(\mathsf{log1p}\left({\color{blue}{\left(e^{x}\right)}}^{-2}\right) - \log 2\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      19. lower-log.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\left(\mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right) - \color{blue}{\log 2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      20. metadata-eval 100.0

          \[\leadsto \mathsf{expm1}\left(\left(\mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right) - \log 2\right) \cdot \color{blue}{-1}\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right) - \log 2\right) \cdot -1\right)} \]

   if -2 < (*.f64 #s(literal -2 binary64) x) < 5.0000000000000001e-4

   1. Initial program 7.7%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)} + 1\right) \]

      3. metadata-eval N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}\right) + 1\right) \]

      4. distribute-lft-in N/A

         \[\leadsto x \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{-1}{3}\right)} + 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{3} \cdot {x}^{2}}\right) + 1\right) \]

      6. associate-+l+ N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \left(\frac{-1}{3} \cdot {x}^{2} + 1\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\left(1 + \frac{-1}{3} \cdot {x}^{2}\right)}\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)\right) + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{2}{15}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \frac{2}{15}} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}, \frac{2}{15}, x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      13. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

   if 5.0000000000000001e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -2:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)\\ \mathbf{elif}\;x \cdot -2 \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{e^{x \cdot -2} + 1} - 1\\ \mathbf{if}\;x \cdot -2 \leq -2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot -2 \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 (+ (exp (* x -2.0)) 1.0)) 1.0)))
   (if (<= (* x -2.0) -2.0)
     t_0
     (if (<= (* x -2.0) 0.0005)
       (fma
        (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x x))
        x
        x)
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (exp((x * -2.0)) + 1.0)) - 1.0;
	double tmp;
	if ((x * -2.0) <= -2.0) {
		tmp = t_0;
	} else if ((x * -2.0) <= 0.0005) {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * (x * x)), x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(exp(Float64(x * -2.0)) + 1.0)) - 1.0)
	tmp = 0.0
	if (Float64(x * -2.0) <= -2.0)
		tmp = t_0;
	elseif (Float64(x * -2.0) <= 0.0005)
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * Float64(x * x)), x, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 / N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[N[(x * -2.0), $MachinePrecision], -2.0], t$95$0, If[LessEqual[N[(x * -2.0), $MachinePrecision], 0.0005], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -2 or 5.0000000000000001e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   if -2 < (*.f64 #s(literal -2 binary64) x) < 5.0000000000000001e-4

   1. Initial program 7.7%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)} + 1\right) \]

      3. metadata-eval N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}\right) + 1\right) \]

      4. distribute-lft-in N/A

         \[\leadsto x \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{-1}{3}\right)} + 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{3} \cdot {x}^{2}}\right) + 1\right) \]

      6. associate-+l+ N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \left(\frac{-1}{3} \cdot {x}^{2} + 1\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\left(1 + \frac{-1}{3} \cdot {x}^{2}\right)}\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)\right) + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{2}{15}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \frac{2}{15}} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}, \frac{2}{15}, x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      13. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -2:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \mathbf{elif}\;x \cdot -2 \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x \cdot -2} + 1} \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp (* x -2.0)) 1.0)) 5e-91)
   (- (/ -1.0 (- x 1.0)) 1.0)
   (fma (* -0.3333333333333333 (* x x)) x x)))

C

double code(double x, double y) {
	double tmp;
	if ((2.0 / (exp((x * -2.0)) + 1.0)) <= 5e-91) {
		tmp = (-1.0 / (x - 1.0)) - 1.0;
	} else {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(Float64(x * -2.0)) + 1.0)) <= 5e-91)
		tmp = Float64(Float64(-1.0 / Float64(x - 1.0)) - 1.0);
	else
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(2.0 / N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 5e-91], N[(N[(-1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) < 4.99999999999999997e-91

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 5.8

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

   5. Applied rewrites 5.8%

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

   7. Applied rewrites 5.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 95.9%

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

   if 4.99999999999999997e-91 < (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))))

   1. Initial program 39.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)} + 1\right) \]

      3. metadata-eval N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}\right) + 1\right) \]

      4. distribute-lft-in N/A

         \[\leadsto x \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{-1}{3}\right)} + 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{3} \cdot {x}^{2}}\right) + 1\right) \]

      6. associate-+l+ N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \left(\frac{-1}{3} \cdot {x}^{2} + 1\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\left(1 + \frac{-1}{3} \cdot {x}^{2}\right)}\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)\right) + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{2}{15}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \frac{2}{15}} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}, \frac{2}{15}, x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)\right)} \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      13. lower-*.f64 66.6

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   8. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 65.4%

       \[\leadsto \mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x \cdot -2} + 1} \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x \cdot -2} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, x, -2\right) \cdot x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* x -2.0)) 2.0)
   (fma (* -0.3333333333333333 (* x x)) x x)
   (- (/ 2.0 (* (fma 2.0 x -2.0) x)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (exp((x * -2.0)) <= 2.0) {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	} else {
		tmp = (2.0 / (fma(2.0, x, -2.0) * x)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(x * -2.0)) <= 2.0)
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	else
		tmp = Float64(Float64(2.0 / Float64(fma(2.0, x, -2.0) * x)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(2.0 / N[(N[(2.0 * x + -2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 #s(literal -2 binary64) x)) < 2

   1. Initial program 39.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)} + 1\right) \]

      3. metadata-eval N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}\right) + 1\right) \]

      4. distribute-lft-in N/A

         \[\leadsto x \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{-1}{3}\right)} + 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{3} \cdot {x}^{2}}\right) + 1\right) \]

      6. associate-+l+ N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \left(\frac{-1}{3} \cdot {x}^{2} + 1\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\left(1 + \frac{-1}{3} \cdot {x}^{2}\right)}\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)\right) + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{2}{15}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \frac{2}{15}} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}, \frac{2}{15}, x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)\right)} \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      13. lower-*.f64 66.6

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   8. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 65.4%

       \[\leadsto \mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right) \]

   if 2 < (exp.f64 (*.f64 #s(literal -2 binary64) x))

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)}} - 1 \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) + 2}} - 1 \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) \cdot x} + 2} - 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2, x, 2\right)}} - 1 \]

      4. sub-neg N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{x \cdot \left(2 + \frac{-4}{3} \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right)}, x, 2\right)} - 1 \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\left(2 + \frac{-4}{3} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(2\right)\right), x, 2\right)} - 1 \]

      6. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\left(2 + \frac{-4}{3} \cdot x\right) \cdot x + \color{blue}{-2}, x, 2\right)} - 1 \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2 + \frac{-4}{3} \cdot x, x, -2\right)}, x, 2\right)} - 1 \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot x + 2}, x, -2\right), x, 2\right)} - 1 \]

      9. lower-fma.f64 98.9

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.3333333333333333, x, 2\right)}, x, -2\right), x, 2\right)} - 1 \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)}} - 1 \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{{x}^{3} \cdot \color{blue}{\left(2 \cdot \frac{1}{x} - \left(\frac{4}{3} + \frac{2}{{x}^{2}}\right)\right)}} - 1 \]

   8. Applied rewrites 98.9%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right) \cdot \color{blue}{x}} - 1 \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\mathsf{fma}\left(2, x, -2\right) \cdot x} - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 98.5%

       \[\leadsto \frac{2}{\mathsf{fma}\left(2, x, -2\right) \cdot x} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x \cdot -2} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, x, -2\right) \cdot x} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x \cdot -2} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot x\right) \cdot x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* x -2.0)) 2.0)
   (fma (* -0.3333333333333333 (* x x)) x x)
   (- (/ 2.0 (* (* 2.0 x) x)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (exp((x * -2.0)) <= 2.0) {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	} else {
		tmp = (2.0 / ((2.0 * x) * x)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(x * -2.0)) <= 2.0)
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(2.0 * x) * x)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(2.0 / N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 #s(literal -2 binary64) x)) < 2

   1. Initial program 39.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)} + 1\right) \]

      3. metadata-eval N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}\right) + 1\right) \]

      4. distribute-lft-in N/A

         \[\leadsto x \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{-1}{3}\right)} + 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{3} \cdot {x}^{2}}\right) + 1\right) \]

      6. associate-+l+ N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \left(\frac{-1}{3} \cdot {x}^{2} + 1\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\left(1 + \frac{-1}{3} \cdot {x}^{2}\right)}\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)\right) + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{2}{15}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \frac{2}{15}} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}, \frac{2}{15}, x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)\right)} \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      13. lower-*.f64 66.6

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   8. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 65.4%

       \[\leadsto \mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right) \]

   if 2 < (exp.f64 (*.f64 #s(literal -2 binary64) x))

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)}} - 1 \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) + 2}} - 1 \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) \cdot x} + 2} - 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2, x, 2\right)}} - 1 \]

      4. sub-neg N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{x \cdot \left(2 + \frac{-4}{3} \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right)}, x, 2\right)} - 1 \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\left(2 + \frac{-4}{3} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(2\right)\right), x, 2\right)} - 1 \]

      6. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\left(2 + \frac{-4}{3} \cdot x\right) \cdot x + \color{blue}{-2}, x, 2\right)} - 1 \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2 + \frac{-4}{3} \cdot x, x, -2\right)}, x, 2\right)} - 1 \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot x + 2}, x, -2\right), x, 2\right)} - 1 \]

      9. lower-fma.f64 98.9

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.3333333333333333, x, 2\right)}, x, -2\right), x, 2\right)} - 1 \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)}} - 1 \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{{x}^{3} \cdot \color{blue}{\left(2 \cdot \frac{1}{x} - \frac{4}{3}\right)}} - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 98.9%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right) \cdot x\right) \cdot \color{blue}{x}} - 1 \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\left(2 \cdot x\right) \cdot x} - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 98.5%

       \[\leadsto \frac{2}{\left(2 \cdot x\right) \cdot x} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x \cdot -2} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 \cdot x\right) \cdot x} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) 0.0005)
   (fma (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x x)) x x)
   (- (/ 2.0 (fma (fma (fma -1.3333333333333333 x 2.0) x -2.0) x 2.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= 0.0005) {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * (x * x)), x, x);
	} else {
		tmp = (2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= 0.0005)
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * Float64(x * x)), x, x);
	else
		tmp = Float64(Float64(2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], 0.0005], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(-1.3333333333333333 * x + 2.0), $MachinePrecision] * x + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < 5.0000000000000001e-4

   1. Initial program 39.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)} + 1\right) \]

      3. metadata-eval N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}\right) + 1\right) \]

      4. distribute-lft-in N/A

         \[\leadsto x \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{-1}{3}\right)} + 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{3} \cdot {x}^{2}}\right) + 1\right) \]

      6. associate-+l+ N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \left(\frac{-1}{3} \cdot {x}^{2} + 1\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\left(1 + \frac{-1}{3} \cdot {x}^{2}\right)}\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)\right) + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{2}{15}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \frac{2}{15}} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}, \frac{2}{15}, x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)\right)} \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      13. lower-*.f64 66.6

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   8. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

   if 5.0000000000000001e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)}} - 1 \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) + 2}} - 1 \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) \cdot x} + 2} - 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2, x, 2\right)}} - 1 \]

      4. sub-neg N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{x \cdot \left(2 + \frac{-4}{3} \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right)}, x, 2\right)} - 1 \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\left(2 + \frac{-4}{3} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(2\right)\right), x, 2\right)} - 1 \]

      6. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\left(2 + \frac{-4}{3} \cdot x\right) \cdot x + \color{blue}{-2}, x, 2\right)} - 1 \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2 + \frac{-4}{3} \cdot x, x, -2\right)}, x, 2\right)} - 1 \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot x + 2}, x, -2\right), x, 2\right)} - 1 \]

      9. lower-fma.f64 98.9

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.3333333333333333, x, 2\right)}, x, -2\right), x, 2\right)} - 1 \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)}} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.16:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.16)
   (- (/ 2.0 (* (fma (fma -1.3333333333333333 x 2.0) x -2.0) x)) 1.0)
   (fma
    (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.16) {
		tmp = (2.0 / (fma(fma(-1.3333333333333333, x, 2.0), x, -2.0) * x)) - 1.0;
	} else {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.16)
		tmp = Float64(Float64(2.0 / Float64(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0) * x)) - 1.0);
	else
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.16], N[(N[(2.0 / N[(N[(N[(-1.3333333333333333 * x + 2.0), $MachinePrecision] * x + -2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15999999999999992

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)}} - 1 \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) + 2}} - 1 \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) \cdot x} + 2} - 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2, x, 2\right)}} - 1 \]

      4. sub-neg N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{x \cdot \left(2 + \frac{-4}{3} \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right)}, x, 2\right)} - 1 \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\left(2 + \frac{-4}{3} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(2\right)\right), x, 2\right)} - 1 \]

      6. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\left(2 + \frac{-4}{3} \cdot x\right) \cdot x + \color{blue}{-2}, x, 2\right)} - 1 \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2 + \frac{-4}{3} \cdot x, x, -2\right)}, x, 2\right)} - 1 \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot x + 2}, x, -2\right), x, 2\right)} - 1 \]

      9. lower-fma.f64 98.9

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.3333333333333333, x, 2\right)}, x, -2\right), x, 2\right)} - 1 \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)}} - 1 \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{{x}^{3} \cdot \color{blue}{\left(2 \cdot \frac{1}{x} - \left(\frac{4}{3} + \frac{2}{{x}^{2}}\right)\right)}} - 1 \]

   8. Applied rewrites 98.9%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right) \cdot \color{blue}{x}} - 1 \]

   if -1.15999999999999992 < x

   1. Initial program 39.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)} + 1\right) \]

      3. metadata-eval N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}\right) + 1\right) \]

      4. distribute-lft-in N/A

         \[\leadsto x \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{-1}{3}\right)} + 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{3} \cdot {x}^{2}}\right) + 1\right) \]

      6. associate-+l+ N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \left(\frac{-1}{3} \cdot {x}^{2} + 1\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\left(1 + \frac{-1}{3} \cdot {x}^{2}\right)}\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)\right) + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{2}{15}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \frac{2}{15}} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}, \frac{2}{15}, x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)\right)} \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      13. lower-*.f64 66.6

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   8. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right) \cdot x\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.32)
   (- (/ 2.0 (* (* (fma -1.3333333333333333 x 2.0) x) x)) 1.0)
   (fma
    (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.32) {
		tmp = (2.0 / ((fma(-1.3333333333333333, x, 2.0) * x) * x)) - 1.0;
	} else {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.32)
		tmp = Float64(Float64(2.0 / Float64(Float64(fma(-1.3333333333333333, x, 2.0) * x) * x)) - 1.0);
	else
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.32], N[(N[(2.0 / N[(N[(N[(-1.3333333333333333 * x + 2.0), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.32000000000000006

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)}} - 1 \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) + 2}} - 1 \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) \cdot x} + 2} - 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2, x, 2\right)}} - 1 \]

      4. sub-neg N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{x \cdot \left(2 + \frac{-4}{3} \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right)}, x, 2\right)} - 1 \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\left(2 + \frac{-4}{3} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(2\right)\right), x, 2\right)} - 1 \]

      6. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\left(2 + \frac{-4}{3} \cdot x\right) \cdot x + \color{blue}{-2}, x, 2\right)} - 1 \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2 + \frac{-4}{3} \cdot x, x, -2\right)}, x, 2\right)} - 1 \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot x + 2}, x, -2\right), x, 2\right)} - 1 \]

      9. lower-fma.f64 98.9

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.3333333333333333, x, 2\right)}, x, -2\right), x, 2\right)} - 1 \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)}} - 1 \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{{x}^{3} \cdot \color{blue}{\left(2 \cdot \frac{1}{x} - \frac{4}{3}\right)}} - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 98.9%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right) \cdot x\right) \cdot \color{blue}{x}} - 1 \]

   if -1.32000000000000006 < x

   1. Initial program 39.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)} + 1\right) \]

      3. metadata-eval N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}\right) + 1\right) \]

      4. distribute-lft-in N/A

         \[\leadsto x \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{-1}{3}\right)} + 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{3} \cdot {x}^{2}}\right) + 1\right) \]

      6. associate-+l+ N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \left(\frac{-1}{3} \cdot {x}^{2} + 1\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\left(1 + \frac{-1}{3} \cdot {x}^{2}\right)}\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)\right) + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{2}{15}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \frac{2}{15}} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}, \frac{2}{15}, x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)\right)} \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      13. lower-*.f64 66.6

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   8. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{2}{\left(\left(-1.3333333333333333 \cdot x\right) \cdot x\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.55)
   (- (/ 2.0 (* (* (* -1.3333333333333333 x) x) x)) 1.0)
   (fma
    (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.55) {
		tmp = (2.0 / (((-1.3333333333333333 * x) * x) * x)) - 1.0;
	} else {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(-1.3333333333333333 * x) * x) * x)) - 1.0);
	else
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.55], N[(N[(2.0 / N[(N[(N[(-1.3333333333333333 * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)}} - 1 \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) + 2}} - 1 \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) \cdot x} + 2} - 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2, x, 2\right)}} - 1 \]

      4. sub-neg N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{x \cdot \left(2 + \frac{-4}{3} \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right)}, x, 2\right)} - 1 \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\left(2 + \frac{-4}{3} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(2\right)\right), x, 2\right)} - 1 \]

      6. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\left(2 + \frac{-4}{3} \cdot x\right) \cdot x + \color{blue}{-2}, x, 2\right)} - 1 \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2 + \frac{-4}{3} \cdot x, x, -2\right)}, x, 2\right)} - 1 \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot x + 2}, x, -2\right), x, 2\right)} - 1 \]

      9. lower-fma.f64 98.9

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.3333333333333333, x, 2\right)}, x, -2\right), x, 2\right)} - 1 \]

   5. Applied rewrites 98.9%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)}} - 1 \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{{x}^{3} \cdot \color{blue}{\left(\left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{{x}^{3}}\right) - \left(\frac{4}{3} + \frac{2}{{x}^{2}}\right)\right)}} - 1 \]

   8. Applied rewrites 98.9%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(-1.3333333333333333, x, 2 - \frac{2 - \frac{2}{x}}{x}\right) \cdot x\right) \cdot \color{blue}{x}} - 1 \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\left(\left(\frac{-4}{3} \cdot x\right) \cdot x\right) \cdot x} - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 98.9%

       \[\leadsto \frac{2}{\left(\left(-1.3333333333333333 \cdot x\right) \cdot x\right) \cdot x} - 1 \]

   if -1.55000000000000004 < x

   1. Initial program 39.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)} + 1\right) \]

      3. metadata-eval N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}\right) + 1\right) \]

      4. distribute-lft-in N/A

         \[\leadsto x \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{-1}{3}\right)} + 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{3} \cdot {x}^{2}}\right) + 1\right) \]

      6. associate-+l+ N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \left(\frac{-1}{3} \cdot {x}^{2} + 1\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\left(1 + \frac{-1}{3} \cdot {x}^{2}\right)}\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)\right) + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{2}{15}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \frac{2}{15}} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}, \frac{2}{15}, x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)\right)} \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      13. lower-*.f64 66.6

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   8. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 75.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.16:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, -2\right), x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.16)
   (- (/ 2.0 (fma (fma 2.0 x -2.0) x 2.0)) 1.0)
   (fma
    (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.16) {
		tmp = (2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0;
	} else {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.16)
		tmp = Float64(Float64(2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0);
	else
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.16], N[(N[(2.0 / N[(N[(2.0 * x + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15999999999999992

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{2 \cdot x + \left(\mathsf{neg}\left(2\right)\right)}, x, 2\right)} - 1 \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(2 \cdot x + \color{blue}{-2}, x, 2\right)} - 1 \]

      6. lower-fma.f64 98.5

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2, x, -2\right)}, x, 2\right)} - 1 \]

   5. Applied rewrites 98.5%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, -2\right), x, 2\right)}} - 1 \]

   if -1.15999999999999992 < x

   1. Initial program 39.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)} + 1\right) \]

      3. metadata-eval N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}\right) + 1\right) \]

      4. distribute-lft-in N/A

         \[\leadsto x \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{-1}{3}\right)} + 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{3} \cdot {x}^{2}}\right) + 1\right) \]

      6. associate-+l+ N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \left(\frac{-1}{3} \cdot {x}^{2} + 1\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\left(1 + \frac{-1}{3} \cdot {x}^{2}\right)}\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)\right) + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{2}{15}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \frac{2}{15}} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}, \frac{2}{15}, x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)\right)} \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      13. lower-*.f64 66.6

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   8. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 74.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, -2\right), x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (- (/ 2.0 (fma (fma 2.0 x -2.0) x 2.0)) 1.0)
   (fma (* -0.3333333333333333 (* x x)) x x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0;
	} else {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0);
	else
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(2.0 / N[(N[(2.0 * x + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{2 \cdot x + \left(\mathsf{neg}\left(2\right)\right)}, x, 2\right)} - 1 \]

      5. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(2 \cdot x + \color{blue}{-2}, x, 2\right)} - 1 \]

      6. lower-fma.f64 98.5

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2, x, -2\right)}, x, 2\right)} - 1 \]

   5. Applied rewrites 98.5%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, -2\right), x, 2\right)}} - 1 \]

   if -1 < x

   1. Initial program 39.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. sub-neg N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)} + 1\right) \]

      3. metadata-eval N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}\right) + 1\right) \]

      4. distribute-lft-in N/A

         \[\leadsto x \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{-1}{3}\right)} + 1\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \left(\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{3} \cdot {x}^{2}}\right) + 1\right) \]

      6. associate-+l+ N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \left(\frac{-1}{3} \cdot {x}^{2} + 1\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\left(1 + \frac{-1}{3} \cdot {x}^{2}\right)}\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)\right) + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      10. *-commutative N/A

          \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{2}{15}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \frac{2}{15}} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}, \frac{2}{15}, x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)\right)} \]

   5. Applied rewrites 65.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      13. lower-*.f64 66.6

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   8. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 65.4%

       \[\leadsto \mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 50.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (* -0.3333333333333333 (* x x)) x x))

C

double code(double x, double y) {
	return fma((-0.3333333333333333 * (x * x)), x, x);
}

Julia

function code(x, y)
	return fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x)
end

Wolfram

code[x_, y_] := N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 51.3%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

   2. sub-neg N/A

      \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)} + 1\right) \]

   3. metadata-eval N/A

      \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}\right) + 1\right) \]

   4. distribute-lft-in N/A

      \[\leadsto x \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{-1}{3}\right)} + 1\right) \]

   5. *-commutative N/A

      \[\leadsto x \cdot \left(\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{3} \cdot {x}^{2}}\right) + 1\right) \]

   6. associate-+l+ N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \left(\frac{-1}{3} \cdot {x}^{2} + 1\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right) + \color{blue}{\left(1 + \frac{-1}{3} \cdot {x}^{2}\right)}\right) \]

   8. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)\right) + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]

   9. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

   10. *-commutative N/A

       \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left({x}^{2} \cdot \frac{2}{15}\right)} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

   11. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}\right) \cdot \frac{2}{15}} + x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot {x}^{2}\right) \cdot {x}^{2}, \frac{2}{15}, x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)\right)} \]

5. Applied rewrites 53.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, 0.13333333333333333, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

   4. unpow2 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

   5. cube-mult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + x \cdot 1 \]

   6. *-rgt-identity N/A

      \[\leadsto {x}^{3} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

   13. lower-*.f64 54.6

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

8. Applied rewrites 54.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
9. Step-by-step derivation

10. Applied rewrites 54.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

11. Taylor expanded in x around 0

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

13. Applied rewrites 53.1%

    \[\leadsto \mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right) \]
14. Add Preprocessing

Alternative 13: 6.6% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \left(1 + x\right) - 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (+ 1.0 x) 1.0))

C

double code(double x, double y) {
	return (1.0 + x) - 1.0;
}

Fortran

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

Java

public static double code(double x, double y) {
	return (1.0 + x) - 1.0;
}

Python

def code(x, y):
	return (1.0 + x) - 1.0

Julia

function code(x, y)
	return Float64(Float64(1.0 + x) - 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 + x) - 1.0;
end

Wolfram

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

Derivation

1. Initial program 51.3%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. lower-+.f64 6.6

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

5. Applied rewrites 6.6%

   \[\leadsto \color{blue}{\left(1 + x\right)} - 1 \]
6. Add Preprocessing

Alternative 14: 4.2% accurate, 30.8× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 1.0))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0 - 1.0;
}

Python

def code(x, y):
	return 1.0 - 1.0

Julia

function code(x, y)
	return Float64(1.0 - 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - 1.0;
end

Wolfram

code[x_, y_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 51.3%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.3%

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

Reproduce

herbie shell --seed 2024284 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))