Logistic function from Lakshay Garg

Percentage Accurate: 53.9% → 99.1%

Time: 5.7s

Alternatives: 11

Speedup: 5.1×

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: 53.9% 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: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -0.5:\\ \;\;\;\;e^{-\log \left({\left(-1 - \frac{-2}{1 + {\left(e^{x}\right)}^{-2}}\right)}^{-1}\right)}\\ \mathbf{elif}\;x \cdot -2 \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) -0.5)
   (exp (- (log (pow (- -1.0 (/ -2.0 (+ 1.0 (pow (exp x) -2.0)))) -1.0))))
   (if (<= (* x -2.0) 1e-5)
     (fma
      (*
       (fma
        (fma -0.05396825396825397 (* x x) 0.13333333333333333)
        (* x x)
        -0.3333333333333333)
       (* x x))
      x
      x)
     (- (/ -1.0 (- x 1.0)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= -0.5) {
		tmp = exp(-log(pow((-1.0 - (-2.0 / (1.0 + pow(exp(x), -2.0)))), -1.0)));
	} else if ((x * -2.0) <= 1e-5) {
		tmp = fma((fma(fma(-0.05396825396825397, (x * x), 0.13333333333333333), (x * x), -0.3333333333333333) * (x * x)), x, x);
	} else {
		tmp = (-1.0 / (x - 1.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= -0.5)
		tmp = exp(Float64(-log((Float64(-1.0 - Float64(-2.0 / Float64(1.0 + (exp(x) ^ -2.0)))) ^ -1.0))));
	elseif (Float64(x * -2.0) <= 1e-5)
		tmp = fma(Float64(fma(fma(-0.05396825396825397, Float64(x * x), 0.13333333333333333), Float64(x * x), -0.3333333333333333) * Float64(x * x)), x, x);
	else
		tmp = Float64(Float64(-1.0 / Float64(x - 1.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], -0.5], N[Exp[(-N[Log[N[Power[N[(-1.0 - N[(-2.0 / N[(1.0 + N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision])], $MachinePrecision], If[LessEqual[N[(x * -2.0), $MachinePrecision], 1e-5], N[(N[(N[(N[(-0.05396825396825397 * N[(x * x), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(-1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.5

   1. Initial program 99.9%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. flip3-- N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-to-exp N/A

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

      7. lower-exp.f64 N/A

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

   4. Applied rewrites 100.0%

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

   if -0.5 < (*.f64 #s(literal -2 binary64) x) < 1.00000000000000008e-5

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 99.7

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

   5. Applied rewrites 99.7%

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

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

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

   if 1.00000000000000008e-5 < (*.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

      \[\leadsto \color{blue}{1} - 1 \]
   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 5.1

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

   8. Applied rewrites 5.1%

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

   10. Applied rewrites 4.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 100.0%

       \[\leadsto \frac{-1}{\color{blue}{x} - 1} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -0.5:\\ \;\;\;\;e^{-\log \left({\left(-1 - \frac{-2}{1 + {\left(e^{x}\right)}^{-2}}\right)}^{-1}\right)}\\ \mathbf{elif}\;x \cdot -2 \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -0.5:\\ \;\;\;\;\frac{1}{{\left(-1 - \frac{-2}{e^{x \cdot -2} + 1}\right)}^{-1}}\\ \mathbf{elif}\;x \cdot -2 \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) -0.5)
   (/ 1.0 (pow (- -1.0 (/ -2.0 (+ (exp (* x -2.0)) 1.0))) -1.0))
   (if (<= (* x -2.0) 1e-5)
     (fma
      (*
       (fma
        (fma -0.05396825396825397 (* x x) 0.13333333333333333)
        (* x x)
        -0.3333333333333333)
       (* x x))
      x
      x)
     (- (/ -1.0 (- x 1.0)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= -0.5) {
		tmp = 1.0 / pow((-1.0 - (-2.0 / (exp((x * -2.0)) + 1.0))), -1.0);
	} else if ((x * -2.0) <= 1e-5) {
		tmp = fma((fma(fma(-0.05396825396825397, (x * x), 0.13333333333333333), (x * x), -0.3333333333333333) * (x * x)), x, x);
	} else {
		tmp = (-1.0 / (x - 1.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= -0.5)
		tmp = Float64(1.0 / (Float64(-1.0 - Float64(-2.0 / Float64(exp(Float64(x * -2.0)) + 1.0))) ^ -1.0));
	elseif (Float64(x * -2.0) <= 1e-5)
		tmp = fma(Float64(fma(fma(-0.05396825396825397, Float64(x * x), 0.13333333333333333), Float64(x * x), -0.3333333333333333) * Float64(x * x)), x, x);
	else
		tmp = Float64(Float64(-1.0 / Float64(x - 1.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], -0.5], N[(1.0 / N[Power[N[(-1.0 - N[(-2.0 / N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * -2.0), $MachinePrecision], 1e-5], N[(N[(N[(N[(-0.05396825396825397 * N[(x * x), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(-1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.5

   1. Initial program 99.9%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. pow-exp N/A

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

      4. *-commutative N/A

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

      5. lower-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if -0.5 < (*.f64 #s(literal -2 binary64) x) < 1.00000000000000008e-5

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 99.7

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

   5. Applied rewrites 99.7%

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

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

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

   if 1.00000000000000008e-5 < (*.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

      \[\leadsto \color{blue}{1} - 1 \]
   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 5.1

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

   8. Applied rewrites 5.1%

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

   10. Applied rewrites 4.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 100.0%

       \[\leadsto \frac{-1}{\color{blue}{x} - 1} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -0.5:\\ \;\;\;\;\frac{1}{{\left(-1 - \frac{-2}{e^{x \cdot -2} + 1}\right)}^{-1}}\\ \mathbf{elif}\;x \cdot -2 \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(-4 \cdot x\right)}, \mathsf{expm1}\left(x \cdot -2\right), -1\right)\\ \mathbf{elif}\;x \cdot -2 \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) -0.04)
   (fma (/ 2.0 (expm1 (* -4.0 x))) (expm1 (* x -2.0)) -1.0)
   (if (<= (* x -2.0) 1e-5)
     (fma (* (* x x) x) -0.3333333333333333 x)
     (- (/ -1.0 (- x 1.0)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= -0.04) {
		tmp = fma((2.0 / expm1((-4.0 * x))), expm1((x * -2.0)), -1.0);
	} else if ((x * -2.0) <= 1e-5) {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	} else {
		tmp = (-1.0 / (x - 1.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= -0.04)
		tmp = fma(Float64(2.0 / expm1(Float64(-4.0 * x))), expm1(Float64(x * -2.0)), -1.0);
	elseif (Float64(x * -2.0) <= 1e-5)
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	else
		tmp = Float64(Float64(-1.0 / Float64(x - 1.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], -0.04], N[(N[(2.0 / N[(Exp[N[(-4.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[(Exp[N[(x * -2.0), $MachinePrecision]] - 1), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * -2.0), $MachinePrecision], 1e-5], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], N[(N[(-1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.0400000000000000008

   1. Initial program 99.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(-4 \cdot x\right)}, \mathsf{expm1}\left(x \cdot -2\right), -1\right)} \]

   if -0.0400000000000000008 < (*.f64 #s(literal -2 binary64) x) < 1.00000000000000008e-5

   1. Initial program 7.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   if 1.00000000000000008e-5 < (*.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

      \[\leadsto \color{blue}{1} - 1 \]
   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 5.1

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

   8. Applied rewrites 5.1%

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

   10. Applied rewrites 4.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 100.0%

       \[\leadsto \frac{-1}{\color{blue}{x} - 1} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(-4 \cdot x\right)}, \mathsf{expm1}\left(x \cdot -2\right), -1\right)\\ \mathbf{elif}\;x \cdot -2 \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -0.5:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \mathbf{elif}\;x \cdot -2 \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) -0.5)
   (- (/ 2.0 (+ (exp (* x -2.0)) 1.0)) 1.0)
   (if (<= (* x -2.0) 1e-5)
     (fma
      (*
       (fma
        (fma -0.05396825396825397 (* x x) 0.13333333333333333)
        (* x x)
        -0.3333333333333333)
       (* x x))
      x
      x)
     (- (/ -1.0 (- x 1.0)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= -0.5) {
		tmp = (2.0 / (exp((x * -2.0)) + 1.0)) - 1.0;
	} else if ((x * -2.0) <= 1e-5) {
		tmp = fma((fma(fma(-0.05396825396825397, (x * x), 0.13333333333333333), (x * x), -0.3333333333333333) * (x * x)), x, x);
	} else {
		tmp = (-1.0 / (x - 1.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= -0.5)
		tmp = Float64(Float64(2.0 / Float64(exp(Float64(x * -2.0)) + 1.0)) - 1.0);
	elseif (Float64(x * -2.0) <= 1e-5)
		tmp = fma(Float64(fma(fma(-0.05396825396825397, Float64(x * x), 0.13333333333333333), Float64(x * x), -0.3333333333333333) * Float64(x * x)), x, x);
	else
		tmp = Float64(Float64(-1.0 / Float64(x - 1.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], -0.5], 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], 1e-5], N[(N[(N[(N[(-0.05396825396825397 * N[(x * x), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(-1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.5

   1. Initial program 99.9%

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

   if -0.5 < (*.f64 #s(literal -2 binary64) x) < 1.00000000000000008e-5

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 99.7

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

   5. Applied rewrites 99.7%

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

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

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

   if 1.00000000000000008e-5 < (*.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

      \[\leadsto \color{blue}{1} - 1 \]
   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 5.1

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

   8. Applied rewrites 5.1%

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

   10. Applied rewrites 4.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 100.0%

       \[\leadsto \frac{-1}{\color{blue}{x} - 1} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -0.5:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \mathbf{elif}\;x \cdot -2 \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) 1e-5)
   (/ 1.0 (/ (fma 0.3333333333333333 (* x x) 1.0) x))
   (- (/ -1.0 (- x 1.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= 1e-5) {
		tmp = 1.0 / (fma(0.3333333333333333, (x * x), 1.0) / x);
	} else {
		tmp = (-1.0 / (x - 1.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= 1e-5)
		tmp = Float64(1.0 / Float64(fma(0.3333333333333333, Float64(x * x), 1.0) / x));
	else
		tmp = Float64(Float64(-1.0 / Float64(x - 1.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], 1e-5], N[(1.0 / N[(N[(0.3333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < 1.00000000000000008e-5

   1. Initial program 36.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

   4. Applied rewrites 36.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + \frac{1}{3} \cdot {x}^{2}}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1 + \frac{1}{3} \cdot {x}^{2}}{x}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{3} \cdot {x}^{2} + 1}}{x}} \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.3

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(0.3333333333333333, \color{blue}{x \cdot x}, 1\right)}{x}} \]

   7. Applied rewrites 70.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, x \cdot x, 1\right)}{x}}} \]

   if 1.00000000000000008e-5 < (*.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

      \[\leadsto \color{blue}{1} - 1 \]
   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 5.1

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

   8. Applied rewrites 5.1%

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

   10. Applied rewrites 4.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 100.0%

       \[\leadsto \frac{-1}{\color{blue}{x} - 1} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

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

Alternative 6: 75.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.5)
   (- (/ -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 (x <= -1.5) {
		tmp = (-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 (x <= -1.5)
		tmp = Float64(Float64(-1.0 / Float64(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[x, -1.5], N[(N[(-1.0 / N[(x - 1.0), $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 < -1.5

   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 5.1

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

   8. Applied rewrites 5.1%

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

   10. Applied rewrites 4.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 100.0%

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

   if -1.5 < x

   1. Initial program 36.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 69.0

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

   5. Applied rewrites 69.0%

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

   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 70.0

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

   8. Applied rewrites 70.0%

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

   10. Applied rewrites 70.0%

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

Alternative 7: 75.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35) (- (/ -1.0 (- x 1.0)) 1.0) (/ 1.0 (/ 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35) {
		tmp = (-1.0 / (x - 1.0)) - 1.0;
	} else {
		tmp = 1.0 / (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.35d0)) then
        tmp = ((-1.0d0) / (x - 1.0d0)) - 1.0d0
    else
        tmp = 1.0d0 / (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35) {
		tmp = (-1.0 / (x - 1.0)) - 1.0;
	} else {
		tmp = 1.0 / (1.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.35:
		tmp = (-1.0 / (x - 1.0)) - 1.0
	else:
		tmp = 1.0 / (1.0 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35)
		tmp = Float64(Float64(-1.0 / Float64(x - 1.0)) - 1.0);
	else
		tmp = Float64(1.0 / Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35)
		tmp = (-1.0 / (x - 1.0)) - 1.0;
	else
		tmp = 1.0 / (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35], N[(N[(-1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3500000000000001

   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 5.1

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

   8. Applied rewrites 5.1%

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

   10. Applied rewrites 4.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 100.0%

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

   if -1.3500000000000001 < x

   1. Initial program 36.8%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-- N/A

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

      7. lift--.f64 N/A

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

      8. inv-pow N/A

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

      9. metadata-eval N/A

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

   4. Applied rewrites 36.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 70.0

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

   7. Applied rewrites 70.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.5% accurate, 5.1× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.3) {
		tmp = (-1.0 / (x - 1.0)) - 1.0;
	} else {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(Float64(-1.0 / Float64(x - 1.0)) - 1.0);
	else
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.3], N[(N[(-1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + 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 5.1

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

   8. Applied rewrites 5.1%

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

   10. Applied rewrites 4.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 100.0%

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

   if -1.30000000000000004 < x

   1. Initial program 36.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 69.0

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

   5. Applied rewrites 69.0%

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

   7. Applied rewrites 69.0%

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

Alternative 9: 50.5% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. pow-plus N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 54.6

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

5. Applied rewrites 54.6%

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

7. Applied rewrites 54.6%

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

Alternative 10: 6.5% 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 50.1%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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 6.8

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

5. Applied rewrites 6.8%

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

Alternative 11: 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 50.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.5%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))