Logistic function from Lakshay Garg

Percentage Accurate: 54.0% → 99.2%

Time: 8.3s

Alternatives: 6

Speedup: 4.2×

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: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -20:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \mathbf{elif}\;x \cdot -2 \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(-1.3333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) -20.0)
   (- (/ 2.0 (+ (exp (* x -2.0)) 1.0)) 1.0)
   (if (<= (* x -2.0) 0.0001)
     (*
      (fma (fma 0.13333333333333333 (* x x) -0.3333333333333333) (* x x) 1.0)
      x)
     (- (/ 2.0 (* (* -1.3333333333333333 (* x x)) x)) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= -20.0) {
		tmp = (2.0 / (exp((x * -2.0)) + 1.0)) - 1.0;
	} else if ((x * -2.0) <= 0.0001) {
		tmp = fma(fma(0.13333333333333333, (x * x), -0.3333333333333333), (x * x), 1.0) * x;
	} else {
		tmp = (2.0 / ((-1.3333333333333333 * (x * x)) * x)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= -20.0)
		tmp = Float64(Float64(2.0 / Float64(exp(Float64(x * -2.0)) + 1.0)) - 1.0);
	elseif (Float64(x * -2.0) <= 0.0001)
		tmp = Float64(fma(fma(0.13333333333333333, Float64(x * x), -0.3333333333333333), Float64(x * x), 1.0) * x);
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(-1.3333333333333333 * Float64(x * x)) * x)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], -20.0], N[(N[(2.0 / N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[N[(x * -2.0), $MachinePrecision], 0.0001], N[(N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(2.0 / N[(N[(-1.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -20

   1. Initial program 100.0%

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

   if -20 < (*.f64 #s(literal -2 binary64) x) < 1.00000000000000005e-4

   1. Initial program 9.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]

   if 1.00000000000000005e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -20:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \mathbf{elif}\;x \cdot -2 \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(-1.3333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) 2e-7)
   x
   (- (/ 2.0 (fma (fma (fma -1.3333333333333333 x 2.0) x -2.0) x 2.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= 2e-7) {
		tmp = x;
	} else {
		tmp = (2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= 2e-7)
		tmp = x;
	else
		tmp = Float64(Float64(2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], 2e-7], x, N[(N[(2.0 / N[(N[(N[(-1.3333333333333333 * x + 2.0), $MachinePrecision] * x + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < 1.9999999999999999e-7

   1. Initial program 43.1%

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

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. sqr-pow N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 42.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(0.5 \cdot \left(e^{x \cdot -2} + 1\right)\right)}^{-0.5}, {\left(0.5 \cdot \left(e^{x \cdot -2} + 1\right)\right)}^{-0.5}, -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

      16. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      17. +-lft-identity 64.1

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

   7. Applied rewrites 64.1%

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

   if 1.9999999999999999e-7 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.6%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq 0.0001:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(-1.3333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) 0.0001)
   x
   (- (/ 2.0 (* (* -1.3333333333333333 (* x x)) x)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= 0.0001) {
		tmp = x;
	} else {
		tmp = (2.0 / ((-1.3333333333333333 * (x * x)) * 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 * (-2.0d0)) <= 0.0001d0) then
        tmp = x
    else
        tmp = (2.0d0 / (((-1.3333333333333333d0) * (x * x)) * x)) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= 0.0001) {
		tmp = x;
	} else {
		tmp = (2.0 / ((-1.3333333333333333 * (x * x)) * x)) - 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * -2.0) <= 0.0001:
		tmp = x
	else:
		tmp = (2.0 / ((-1.3333333333333333 * (x * x)) * x)) - 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= 0.0001)
		tmp = x;
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(-1.3333333333333333 * Float64(x * x)) * x)) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * -2.0) <= 0.0001)
		tmp = x;
	else
		tmp = (2.0 / ((-1.3333333333333333 * (x * x)) * x)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], 0.0001], x, N[(N[(2.0 / N[(N[(-1.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < 1.00000000000000005e-4

   1. Initial program 43.3%

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

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. sqr-pow N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(0.5 \cdot \left(e^{x \cdot -2} + 1\right)\right)}^{-0.5}, {\left(0.5 \cdot \left(e^{x \cdot -2} + 1\right)\right)}^{-0.5}, -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

      16. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      17. +-lft-identity 64.1

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

   7. Applied rewrites 64.1%

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

   if 1.00000000000000005e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq 0.0001:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(-1.3333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq 0.0001:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x - 1, x, 1\right)} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) 0.0001) x (- (/ 1.0 (fma (- x 1.0) x 1.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= 0.0001) {
		tmp = x;
	} else {
		tmp = (1.0 / fma((x - 1.0), x, 1.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * -2.0) <= 0.0001)
		tmp = x;
	else
		tmp = Float64(Float64(1.0 / fma(Float64(x - 1.0), x, 1.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], 0.0001], x, N[(N[(1.0 / N[(N[(x - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < 1.00000000000000005e-4

   1. Initial program 43.3%

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

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. sqr-pow N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(0.5 \cdot \left(e^{x \cdot -2} + 1\right)\right)}^{-0.5}, {\left(0.5 \cdot \left(e^{x \cdot -2} + 1\right)\right)}^{-0.5}, -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

      16. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      17. +-lft-identity 64.1

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

   7. Applied rewrites 64.1%

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

   if 1.00000000000000005e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} - 1 \]
   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. metadata-eval N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 5.0

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

   8. Applied rewrites 5.0%

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

   10. Applied rewrites 2.5%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq 0.0001:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x - 1, x, 1\right)} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq 0.0001:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x -2.0) 0.0001) x (- (/ -1.0 (- x 1.0)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= 0.0001) {
		tmp = x;
	} else {
		tmp = (-1.0 / (x - 1.0)) - 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 * (-2.0d0)) <= 0.0001d0) then
        tmp = x
    else
        tmp = ((-1.0d0) / (x - 1.0d0)) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * -2.0) <= 0.0001) {
		tmp = x;
	} else {
		tmp = (-1.0 / (x - 1.0)) - 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * -2.0) <= 0.0001:
		tmp = 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) <= 0.0001)
		tmp = x;
	else
		tmp = Float64(Float64(-1.0 / Float64(x - 1.0)) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * -2.0) <= 0.0001)
		tmp = x;
	else
		tmp = (-1.0 / (x - 1.0)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * -2.0), $MachinePrecision], 0.0001], x, 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) < 1.00000000000000005e-4

   1. Initial program 43.3%

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

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. sqr-pow N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(0.5 \cdot \left(e^{x \cdot -2} + 1\right)\right)}^{-0.5}, {\left(0.5 \cdot \left(e^{x \cdot -2} + 1\right)\right)}^{-0.5}, -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

      16. metadata-eval N/A

          \[\leadsto \color{blue}{0} + x \]

      17. +-lft-identity 64.1

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

   7. Applied rewrites 64.1%

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

   if 1.00000000000000005e-4 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} - 1 \]
   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. metadata-eval N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 5.0

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

   8. Applied rewrites 5.0%

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

   10. Applied rewrites 4.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.7%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq 0.0001:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.5% accurate, 123.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 57.9%

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

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

   5. inv-pow N/A

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

   6. metadata-eval N/A

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

   7. sqr-pow N/A

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

   8. lower-fma.f64 N/A

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

4. Applied rewrites 57.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(0.5 \cdot \left(e^{x \cdot -2} + 1\right)\right)}^{-0.5}, {\left(0.5 \cdot \left(e^{x \cdot -2} + 1\right)\right)}^{-0.5}, -1\right)} \]

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. unpow2 N/A

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

   4. rem-square-sqrt N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. *-lft-identity N/A

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. +-commutative N/A

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

   15. associate-+r+ N/A

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

   16. metadata-eval N/A

       \[\leadsto \color{blue}{0} + x \]

   17. +-lft-identity 48.9

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

7. Applied rewrites 48.9%

   \[\leadsto \color{blue}{x} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))