Logistic function from Lakshay Garg

Percentage Accurate: 53.8% → 99.3%

Time: 6.9s

Alternatives: 9

Speedup: 1.1×

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: 53.8% 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: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000000000 \lor \neg \left(-2 \cdot x \leq 0.05\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -1000000000.0) (not (<= (* -2.0 x) 0.05)))
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (fma
    (pow x 3.0)
    (fma
     (fma -0.05396825396825397 (* x x) 0.13333333333333333)
     (* x x)
     -0.3333333333333333)
    x)))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -1000000000.0) || !((-2.0 * x) <= 0.05)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else {
		tmp = fma(pow(x, 3.0), fma(fma(-0.05396825396825397, (x * x), 0.13333333333333333), (x * x), -0.3333333333333333), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -1000000000.0) || !(Float64(-2.0 * x) <= 0.05))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	else
		tmp = fma((x ^ 3.0), fma(fma(-0.05396825396825397, Float64(x * x), 0.13333333333333333), Float64(x * x), -0.3333333333333333), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -1000000000.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.05]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(-0.05396825396825397 * N[(x * x), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e9 or 0.050000000000000003 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -1e9 < (*.f64 #s(literal -2 binary64) x) < 0.050000000000000003

   1. Initial program 7.9%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 99.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)} + x \cdot 1 \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) + x \cdot 1 \]

      5. cube-mult N/A

         \[\leadsto \color{blue}{{x}^{3}} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) + x \cdot 1 \]

      6. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) + \color{blue}{x} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, {x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, {x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\frac{-17}{315} \cdot {x}^{2} + \frac{2}{15}}, {x}^{2}, \frac{-1}{3}\right), x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-17}{315}, {x}^{2}, \frac{2}{15}\right)}, {x}^{2}, \frac{-1}{3}\right), x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-17}{315}, \color{blue}{x \cdot x}, \frac{2}{15}\right), {x}^{2}, \frac{-1}{3}\right), x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-17}{315}, \color{blue}{x \cdot x}, \frac{2}{15}\right), {x}^{2}, \frac{-1}{3}\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-17}{315}, x \cdot x, \frac{2}{15}\right), \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      18. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000000000 \lor \neg \left(-2 \cdot x \leq 0.05\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, x, 1\right) \cdot \left(1 - x\right)\right)}^{-1} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 5e-29)
   (- (pow (* (fma x x 1.0) (- 1.0 x)) -1.0) 1.0)
   (fma
    (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x, double y) {
	double tmp;
	if ((2.0 / (1.0 + exp((-2.0 * x)))) <= 5e-29) {
		tmp = pow((fma(x, x, 1.0) * (1.0 - x)), -1.0) - 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 (Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) <= 5e-29)
		tmp = Float64((Float64(fma(x, x, 1.0) * Float64(1.0 - x)) ^ -1.0) - 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[N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-29], N[(N[Power[N[(N[(x * x + 1.0), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -1.0], $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 (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) < 4.99999999999999986e-29

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.7

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

   8. Applied rewrites 5.7%

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

   10. Applied rewrites 5.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.0%

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

   if 4.99999999999999986e-29 < (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))))

   1. Initial program 35.2%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 70.6%

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

   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 71.3

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

   8. Applied rewrites 71.3%

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

   10. Applied rewrites 71.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, x, 1\right) \cdot \left(1 - x\right)\right)}^{-1} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x - 1, x, 1\right)\right)}^{-1} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 5e-29)
   (- (pow (fma (- x 1.0) x 1.0) -1.0) 1.0)
   (fma
    (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x, double y) {
	double tmp;
	if ((2.0 / (1.0 + exp((-2.0 * x)))) <= 5e-29) {
		tmp = pow(fma((x - 1.0), x, 1.0), -1.0) - 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 (Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) <= 5e-29)
		tmp = Float64((fma(Float64(x - 1.0), x, 1.0) ^ -1.0) - 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[N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-29], N[(N[Power[N[(N[(x - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], -1.0], $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 (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) < 4.99999999999999986e-29

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.7

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

   8. Applied rewrites 5.7%

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

   10. Applied rewrites 5.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 97.0%

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

   if 4.99999999999999986e-29 < (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))))

   1. Initial program 35.2%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 70.6%

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

   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 71.3

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

   8. Applied rewrites 71.3%

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

   10. Applied rewrites 71.3%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x - 1, x, 1\right)\right)}^{-1} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000000000 \lor \neg \left(-2 \cdot x \leq 0.005\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -1000000000.0) (not (<= (* -2.0 x) 0.005)))
   (- (/ 2.0 (+ 1.0 (exp (* -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 (((-2.0 * x) <= -1000000000.0) || !((-2.0 * x) <= 0.005)) {
		tmp = (2.0 / (1.0 + exp((-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 ((Float64(-2.0 * x) <= -1000000000.0) || !(Float64(-2.0 * x) <= 0.005))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-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[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -1000000000.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.005]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $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 (*.f64 #s(literal -2 binary64) x) < -1e9 or 0.0050000000000000001 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -1e9 < (*.f64 #s(literal -2 binary64) x) < 0.0050000000000000001

   1. Initial program 7.2%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000000000 \lor \neg \left(-2 \cdot x \leq 0.005\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = pow(fma((x - 1.0), x, 1.0), -1.0) - 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((fma(Float64(x - 1.0), x, 1.0) ^ -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[x, -1.0], N[(N[Power[N[(N[(x - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], -1.0], $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

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.7

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

   8. Applied rewrites 5.7%

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

   10. Applied rewrites 5.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 97.0%

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

   if -1 < x

   1. Initial program 35.2%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 70.6%

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

   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 71.3

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

   8. Applied rewrites 71.3%

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

   10. Applied rewrites 71.3%

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

   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 70.3%

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

4. Final simplification 77.6%

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

Alternative 6: 75.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;{\left(1 - x\right)}^{-1} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.3)
   (- (pow (- 1.0 x) -1.0) 1.0)
   (fma (* -0.3333333333333333 (* x x)) x x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.3) {
		tmp = pow((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 (x <= -1.3)
		tmp = Float64((Float64(1.0 - 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[x, -1.3], N[(N[Power[N[(1.0 - x), $MachinePrecision], -1.0], $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 < -1.30000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.7

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

   8. Applied rewrites 5.7%

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

   10. Applied rewrites 5.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 94.9%

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

   if -1.30000000000000004 < x

   1. Initial program 35.2%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 70.6%

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

   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 71.3

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

   8. Applied rewrites 71.3%

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

   10. Applied rewrites 71.3%

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

   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 70.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;{\left(1 - x\right)}^{-1} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.6% 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 52.9%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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 51.8%

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

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 52.9

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

8. Applied rewrites 52.9%

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

10. Applied rewrites 52.9%

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

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 51.3%

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

Alternative 8: 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 52.9%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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 6.3

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

5. Applied rewrites 6.3%

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

Alternative 9: 4.3% 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 52.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.1%

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

Reproduce

herbie shell --seed 2024299 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))