Logistic function from Lakshay Garg

Percentage Accurate: 54.3% → 99.7%

Time: 5.7s

Alternatives: 10

Speedup: 1.2×

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: 54.3% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -0.1:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{fma}\left(-0.5, x, 1\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) -0.1)
   (expm1 (- (log 2.0) (log1p (exp (* x -2.0)))))
   (expm1 (* (fma -0.5 x 1.0) x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= -0.1) {
		tmp = expm1((log(2.0) - log1p(exp((x * -2.0)))));
	} else {
		tmp = expm1((fma(-0.5, x, 1.0) * x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= -0.1)
		tmp = expm1(Float64(log(2.0) - log1p(exp(Float64(x * -2.0)))));
	else
		tmp = expm1(Float64(fma(-0.5, x, 1.0) * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], -0.1], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[(Exp[N[(N[(-0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision]] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.10000000000000001

   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. 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 -0.10000000000000001 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 40.1%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 40.2

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

   4. Applied rewrites 40.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{expm1}\left(\color{blue}{x \cdot \left(1 + \frac{-1}{2} \cdot x\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.5

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.5, x, 1\right)} \cdot x\right) \]

   7. Applied rewrites 99.5%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.5, x, 1\right) \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -0.1:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{fma}\left(-0.5, x, 1\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -0.1:\\ \;\;\;\;\frac{2}{1 + e^{x \cdot -2}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{fma}\left(-0.5, x, 1\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) -0.1)
   (- (/ 2.0 (+ 1.0 (exp (* x -2.0)))) 1.0)
   (expm1 (* (fma -0.5 x 1.0) x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= -0.1) {
		tmp = (2.0 / (1.0 + exp((x * -2.0)))) - 1.0;
	} else {
		tmp = expm1((fma(-0.5, x, 1.0) * x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= -0.1)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(x * -2.0)))) - 1.0);
	else
		tmp = expm1(Float64(fma(-0.5, x, 1.0) * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], -0.1], N[(N[(2.0 / N[(1.0 + N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(Exp[N[(N[(-0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision]] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.10000000000000001

   1. Initial program 99.9%

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

   if -0.10000000000000001 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 40.1%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 40.2

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

   4. Applied rewrites 40.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{expm1}\left(\color{blue}{x \cdot \left(1 + \frac{-1}{2} \cdot x\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.5

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.5, x, 1\right)} \cdot x\right) \]

   7. Applied rewrites 99.5%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.5, x, 1\right) \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -0.1:\\ \;\;\;\;\frac{2}{1 + e^{x \cdot -2}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{fma}\left(-0.5, x, 1\right) \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\mathsf{fma}\left(-0.5, x, 1\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (expm1 (* (fma -0.5 x 1.0) x)))

C

double code(double x, double y) {
	return expm1((fma(-0.5, x, 1.0) * x));
}

Julia

function code(x, y)
	return expm1(Float64(fma(-0.5, x, 1.0) * x))
end

Wolfram

code[x_, y_] := N[(Exp[N[(N[(-0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision]] - 1), $MachinePrecision]

Derivation

1. Initial program 55.8%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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. 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 55.8

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

4. Applied rewrites 55.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{expm1}\left(\color{blue}{x \cdot \left(1 + \frac{-1}{2} \cdot x\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 74.0

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.5, x, 1\right)} \cdot x\right) \]

7. Applied rewrites 74.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.5, x, 1\right) \cdot x}\right) \]
8. Add Preprocessing

Alternative 4: 74.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(-\left(-x\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (expm1 (- (- x))))

C

double code(double x, double y) {
	return expm1(-(-x));
}

Java

public static double code(double x, double y) {
	return Math.expm1(-(-x));
}

Python

def code(x, y):
	return math.expm1(-(-x))

Julia

function code(x, y)
	return expm1(Float64(-Float64(-x)))
end

Wolfram

code[x_, y_] := N[(Exp[(-(-x))] - 1), $MachinePrecision]

Derivation

1. Initial program 55.8%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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. 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 55.8

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

4. Applied rewrites 55.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(-1 \cdot x\right)} \cdot -1\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot -1\right) \]

   2. lower-neg.f64 73.7

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(-x\right)} \cdot -1\right) \]

7. Applied rewrites 73.7%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(-x\right)} \cdot -1\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(-x\right) \cdot -1}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-1 \cdot \left(-x\right)}\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{neg}\left(\left(-x\right)\right)}\right) \]

   4. lower-neg.f64 73.7

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-\left(-x\right)}\right) \]

9. Applied rewrites 73.7%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{-\left(-x\right)}\right) \]
10. Add Preprocessing

Alternative 5: 29.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)} - 1 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (/ 2.0 (fma (fma (fma -1.3333333333333333 x 2.0) x -2.0) x 2.0)) 1.0))

C

double code(double x, double y) {
	return (2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0;
}

Julia

function code(x, y)
	return Float64(Float64(2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0)
end

Wolfram

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

Derivation

1. Initial program 55.8%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 29.7

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

5. Applied rewrites 29.7%

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

Alternative 6: 28.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, -2\right), x, 2\right)} - 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ 2.0 (fma (fma 2.0 x -2.0) x 2.0)) 1.0))

C

double code(double x, double y) {
	return (2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0;
}

Julia

function code(x, y)
	return Float64(Float64(2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0)
end

Wolfram

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

Derivation

1. Initial program 55.8%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 29.4

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

5. Applied rewrites 29.4%

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

Alternative 7: 29.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;\frac{2}{x \cdot -2} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.9) (- (/ 2.0 (* x -2.0)) 1.0) (- (+ 1.0 x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.9) {
		tmp = (2.0 / (x * -2.0)) - 1.0;
	} else {
		tmp = (1.0 + x) - 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.9d0)) then
        tmp = (2.0d0 / (x * (-2.0d0))) - 1.0d0
    else
        tmp = (1.0d0 + x) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.9) {
		tmp = (2.0 / (x * -2.0)) - 1.0;
	} else {
		tmp = (1.0 + x) - 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.9:
		tmp = (2.0 / (x * -2.0)) - 1.0
	else:
		tmp = (1.0 + x) - 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.9)
		tmp = Float64(Float64(2.0 / Float64(x * -2.0)) - 1.0);
	else
		tmp = Float64(Float64(1.0 + x) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.9)
		tmp = (2.0 / (x * -2.0)) - 1.0;
	else
		tmp = (1.0 + x) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.9], N[(N[(2.0 / N[(x * -2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(1.0 + x), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{-2 \cdot \color{blue}{x}} - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 98.0%

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

   if -1.8999999999999999 < x

   1. Initial program 40.7%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 7.4

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

   5. Applied rewrites 7.4%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;\frac{2}{x \cdot -2} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x\right) - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 28.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(-2, x, 2\right)} - 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ 2.0 (fma -2.0 x 2.0)) 1.0))

C

double code(double x, double y) {
	return (2.0 / fma(-2.0, x, 2.0)) - 1.0;
}

Julia

function code(x, y)
	return Float64(Float64(2.0 / fma(-2.0, x, 2.0)) - 1.0)
end

Wolfram

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

Derivation

1. Initial program 55.8%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 29.0

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

5. Applied rewrites 29.0%

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

Alternative 9: 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 55.8%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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 10: 4.2% accurate, 30.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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 2024304 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))