Logistic function from Lakshay Garg

Percentage Accurate: 54.7% → 99.3%

Time: 7.7s

Alternatives: 8

Speedup: 0.9×

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: 54.7% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot -2} + 1\\ t_1 := \frac{2}{t\_0}\\ t_2 := t\_1 - -1\\ \mathbf{if}\;x \cdot -2 \leq -5000:\\ \;\;\;\;t\_1 - 1\\ \mathbf{elif}\;x \cdot -2 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{t\_2} \cdot \left(-1 - \frac{-2}{t\_0}\right)\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (exp (* x -2.0)) 1.0)) (t_1 (/ 2.0 t_0)) (t_2 (- t_1 -1.0)))
   (if (<= (* x -2.0) -5000.0)
     (- t_1 1.0)
     (if (<= (* x -2.0) 2e-13)
       (fma (* (* x x) x) -0.3333333333333333 x)
       (* (* (/ 1.0 t_2) (- -1.0 (/ -2.0 t_0))) t_2)))))

C

double code(double x, double y) {
	double t_0 = exp((x * -2.0)) + 1.0;
	double t_1 = 2.0 / t_0;
	double t_2 = t_1 - -1.0;
	double tmp;
	if ((x * -2.0) <= -5000.0) {
		tmp = t_1 - 1.0;
	} else if ((x * -2.0) <= 2e-13) {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	} else {
		tmp = ((1.0 / t_2) * (-1.0 - (-2.0 / t_0))) * t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(exp(Float64(x * -2.0)) + 1.0)
	t_1 = Float64(2.0 / t_0)
	t_2 = Float64(t_1 - -1.0)
	tmp = 0.0
	if (Float64(x * -2.0) <= -5000.0)
		tmp = Float64(t_1 - 1.0);
	elseif (Float64(x * -2.0) <= 2e-13)
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	else
		tmp = Float64(Float64(Float64(1.0 / t_2) * Float64(-1.0 - Float64(-2.0 / t_0))) * t_2);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - -1.0), $MachinePrecision]}, If[LessEqual[N[(x * -2.0), $MachinePrecision], -5000.0], N[(t$95$1 - 1.0), $MachinePrecision], If[LessEqual[N[(x * -2.0), $MachinePrecision], 2e-13], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], N[(N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(-1.0 - N[(-2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -5e3

   1. Initial program 100.0%

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

   if -5e3 < (*.f64 #s(literal -2 binary64) x) < 2.0000000000000001e-13

   1. Initial program 6.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot x} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(1, \color{blue}{x}, \left(-0.3333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

   if 2.0000000000000001e-13 < (*.f64 #s(literal -2 binary64) x)

   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. flip-- N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. difference-of-sqr-1 N/A

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

      6. lift--.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -5000:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \mathbf{elif}\;x \cdot -2 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{2}{e^{x \cdot -2} + 1} - -1} \cdot \left(-1 - \frac{-2}{e^{x \cdot -2} + 1}\right)\right) \cdot \left(\frac{2}{e^{x \cdot -2} + 1} - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x \cdot -2} + 1} \leq 0.8:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp (* x -2.0)) 1.0)) 0.8)
   (- (/ 2.0 (fma (fma (fma -1.3333333333333333 x 2.0) x -2.0) x 2.0)) 1.0)
   (* 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((2.0 / (exp((x * -2.0)) + 1.0)) <= 0.8) {
		tmp = (2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(Float64(x * -2.0)) + 1.0)) <= 0.8)
		tmp = Float64(Float64(2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(2.0 / N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.8], 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], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) < 0.80000000000000004

   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 97.4

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

   5. Applied rewrites 97.4%

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

   if 0.80000000000000004 < (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))))

   1. Initial program 29.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   5. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 76.9%

      \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x \cdot -2} + 1} \leq 0.8:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x \cdot -2} + 1} \leq 0.8:\\ \;\;\;\;\frac{1}{\left(1 - x\right) \cdot \mathsf{fma}\left(x, x, 1\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp (* x -2.0)) 1.0)) 0.8)
   (- (/ 1.0 (* (- 1.0 x) (fma x x 1.0))) 1.0)
   (* 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((2.0 / (exp((x * -2.0)) + 1.0)) <= 0.8) {
		tmp = (1.0 / ((1.0 - x) * fma(x, x, 1.0))) - 1.0;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(Float64(x * -2.0)) + 1.0)) <= 0.8)
		tmp = Float64(Float64(1.0 / Float64(Float64(1.0 - x) * fma(x, x, 1.0))) - 1.0);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(2.0 / N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.8], N[(N[(1.0 / N[(N[(1.0 - x), $MachinePrecision] * N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) < 0.80000000000000004

   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 6.1

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

   5. Applied rewrites 6.1%

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

   7. Applied rewrites 4.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.4%

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

   if 0.80000000000000004 < (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))))

   1. Initial program 29.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 76.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   5. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 76.9%

      \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x \cdot -2} + 1} \leq 0.8:\\ \;\;\;\;\frac{1}{\left(1 - x\right) \cdot \mathsf{fma}\left(x, x, 1\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x \cdot -2} + 1} \leq 0.01:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, x - 1, 1\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp (* x -2.0)) 1.0)) 0.01)
   (- (/ 1.0 (fma x (- x 1.0) 1.0)) 1.0)
   (* 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((2.0 / (exp((x * -2.0)) + 1.0)) <= 0.01) {
		tmp = (1.0 / fma(x, (x - 1.0), 1.0)) - 1.0;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(Float64(x * -2.0)) + 1.0)) <= 0.01)
		tmp = Float64(Float64(1.0 / fma(x, Float64(x - 1.0), 1.0)) - 1.0);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(2.0 / N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(1.0 / N[(x * N[(x - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) < 0.0100000000000000002

   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 5.8

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

   5. Applied rewrites 5.8%

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.2%

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

   if 0.0100000000000000002 < (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))))

   1. Initial program 29.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 76.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   5. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 76.7%

      \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x \cdot -2} + 1} \leq 0.01:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, x - 1, 1\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{e^{x \cdot -2} + 1} - 1\\ \mathbf{if}\;x \cdot -2 \leq -5000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot -2 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 (+ (exp (* x -2.0)) 1.0)) 1.0)))
   (if (<= (* x -2.0) -5000.0)
     t_0
     (if (<= (* x -2.0) 2e-13)
       (fma (* (* x x) x) -0.3333333333333333 x)
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (exp((x * -2.0)) + 1.0)) - 1.0;
	double tmp;
	if ((x * -2.0) <= -5000.0) {
		tmp = t_0;
	} else if ((x * -2.0) <= 2e-13) {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(exp(Float64(x * -2.0)) + 1.0)) - 1.0)
	tmp = 0.0
	if (Float64(x * -2.0) <= -5000.0)
		tmp = t_0;
	elseif (Float64(x * -2.0) <= 2e-13)
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$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], -5000.0], t$95$0, If[LessEqual[N[(x * -2.0), $MachinePrecision], 2e-13], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -5e3 or 2.0000000000000001e-13 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -5e3 < (*.f64 #s(literal -2 binary64) x) < 2.0000000000000001e-13

   1. Initial program 6.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot x} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(1, \color{blue}{x}, \left(-0.3333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot -2 \leq -5000:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \mathbf{elif}\;x \cdot -2 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{x \cdot -2} + 1} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x \cdot -2} + 1} \leq 0.01:\\ \;\;\;\;\frac{1}{1 - x} - 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp (* x -2.0)) 1.0)) 0.01)
   (- (/ 1.0 (- 1.0 x)) 1.0)
   (* 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((2.0 / (exp((x * -2.0)) + 1.0)) <= 0.01) {
		tmp = (1.0 / (1.0 - x)) - 1.0;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((2.0d0 / (exp((x * (-2.0d0))) + 1.0d0)) <= 0.01d0) then
        tmp = (1.0d0 / (1.0d0 - x)) - 1.0d0
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((2.0 / (Math.exp((x * -2.0)) + 1.0)) <= 0.01) {
		tmp = (1.0 / (1.0 - x)) - 1.0;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (2.0 / (math.exp((x * -2.0)) + 1.0)) <= 0.01:
		tmp = (1.0 / (1.0 - x)) - 1.0
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(Float64(x * -2.0)) + 1.0)) <= 0.01)
		tmp = Float64(Float64(1.0 / Float64(1.0 - x)) - 1.0);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((2.0 / (exp((x * -2.0)) + 1.0)) <= 0.01)
		tmp = (1.0 / (1.0 - x)) - 1.0;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(2.0 / N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) < 0.0100000000000000002

   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 5.8

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

   5. Applied rewrites 5.8%

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

   7. Applied rewrites 5.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.4%

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

   if 0.0100000000000000002 < (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))))

   1. Initial program 29.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 76.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   5. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 76.7%

      \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x \cdot -2} + 1} \leq 0.01:\\ \;\;\;\;\frac{1}{1 - x} - 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.7% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.7%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 59.9

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

5. Applied rewrites 59.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]

6. Taylor expanded in x around 0

   \[\leadsto 1 \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 61.4%

   \[\leadsto 1 \cdot x \]
9. Add Preprocessing

Alternative 8: 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 44.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.5%

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

Reproduce

herbie shell --seed 2024240 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))