Logistic function from Lakshay Garg

Percentage Accurate: 54.4% → 99.3%

Time: 8.1s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -2:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)\right)\\ \mathbf{elif}\;x \cdot -2 \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, x, 1 - x\right)} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) -2.0)
   (expm1 (- (log 2.0) (log1p (exp (* x -2.0)))))
   (if (<= (* x -2.0) 1e-8)
     (fma (* (* x x) x) -0.3333333333333333 x)
     (- (/ 1.0 (fma x x (- 1.0 x))) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= -2.0) {
		tmp = expm1((log(2.0) - log1p(exp((x * -2.0)))));
	} else if ((x * -2.0) <= 1e-8) {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	} else {
		tmp = (1.0 / fma(x, x, (1.0 - x))) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= -2.0)
		tmp = expm1(Float64(log(2.0) - log1p(exp(Float64(x * -2.0)))));
	elseif (Float64(x * -2.0) <= 1e-8)
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	else
		tmp = Float64(Float64(1.0 / fma(x, x, Float64(1.0 - x))) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], -2.0], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[N[(x * -2.0), $MachinePrecision], 1e-8], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], N[(N[(1.0 / N[(x * x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-to-exp N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. log-div N/A

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

      10. lower--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-log1p.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. exp-prod N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-exp.f64 N/A

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

      19. lower-log.f64 N/A

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

      20. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right) - \log 2\right) \cdot -1\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. pow-exp N/A

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

      4. *-commutative N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   if -2 < (*.f64 #s(literal -2 binary64) x) < 1e-8

   1. Initial program 7.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 1e-8 < (*.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

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

      1. lower-+.f64 5.4

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

   5. Applied rewrites 5.4%

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

   7. Applied rewrites 5.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -2:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)\right)\\ \mathbf{elif}\;x \cdot -2 \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, x, 1 - x\right)} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 42.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
   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 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 62.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 5.4

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

   5. Applied rewrites 5.4%

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

   7. Applied rewrites 5.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 72.9%

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

Alternative 3: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -2:\\ \;\;\;\;\frac{2}{1 + e^{x \cdot -2}} - 1\\ \mathbf{elif}\;x \cdot -2 \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, x, 1 - x\right)} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) -2.0)
   (- (/ 2.0 (+ 1.0 (exp (* x -2.0)))) 1.0)
   (if (<= (* x -2.0) 1e-8)
     (fma (* (* x x) x) -0.3333333333333333 x)
     (- (/ 1.0 (fma x x (- 1.0 x))) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= -2.0) {
		tmp = (2.0 / (1.0 + exp((x * -2.0)))) - 1.0;
	} else if ((x * -2.0) <= 1e-8) {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	} else {
		tmp = (1.0 / fma(x, x, (1.0 - x))) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= -2.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(x * -2.0)))) - 1.0);
	elseif (Float64(x * -2.0) <= 1e-8)
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	else
		tmp = Float64(Float64(1.0 / fma(x, x, Float64(1.0 - x))) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], -2.0], N[(N[(2.0 / N[(1.0 + N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[N[(x * -2.0), $MachinePrecision], 1e-8], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], N[(N[(1.0 / N[(x * x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   if -2 < (*.f64 #s(literal -2 binary64) x) < 1e-8

   1. Initial program 7.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 1e-8 < (*.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

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

      1. lower-+.f64 5.4

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

   5. Applied rewrites 5.4%

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

   7. Applied rewrites 5.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -2:\\ \;\;\;\;\frac{2}{1 + e^{x \cdot -2}} - 1\\ \mathbf{elif}\;x \cdot -2 \leq 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, x, 1 - x\right)} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 42.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
   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 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 62.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 5.4

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

   5. Applied rewrites 5.4%

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 4.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 100.0%

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

4. Final simplification 72.9%

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

Alternative 5: 74.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (exp((x * -2.0)) <= 2.0) {
		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 (exp(Float64(x * -2.0)) <= 2.0)
		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[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision], 2.0], 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 2 regimes

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

   1. Initial program 42.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
   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 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 62.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 5.4

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

   5. Applied rewrites 5.4%

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

   7. Applied rewrites 5.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.9%

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

4. Final simplification 72.9%

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

Alternative 6: 49.9% 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 58.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
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 45.4

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

5. Applied rewrites 45.4%

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

7. Applied rewrites 45.4%

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

Alternative 7: 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 58.2%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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 6.1

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

5. Applied rewrites 6.1%

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

Alternative 8: 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 58.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.2%

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

Reproduce

herbie shell --seed 2024277 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))