Logistic function from Lakshay Garg

Percentage Accurate: 54.5% → 99.4%

Time: 9.0s

Alternatives: 10

Speedup: 10.2×

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: 54.5% 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: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -5000.0)
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (if (<= (* -2.0 x) 0.002)
     (fma
      (fma (* x x) 0.13333333333333333 -0.3333333333333333)
      (* x (* x x))
      x)
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5000.0) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.002) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5000.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5000.0], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], -1.0]]

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) < 2e-3

   1. Initial program 8.4%

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

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

   if 2e-3 < (*.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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, -1\right)}{\mathsf{fma}\left(x, x \cdot x, 1\right)}, \mathsf{fma}\left(x, x + -1, 1\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))))
   (if (<= t_0 0.0)
     (fma (/ (fma x x -1.0) (fma x (* x x) 1.0)) (fma x (+ x -1.0) 1.0) 1.0)
     (if (<= t_0 2.0)
       (fma
        (fma (* x x) 0.13333333333333333 -0.3333333333333333)
        (* x (* x x))
        x)
       -1.0))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fma((fma(x, x, -1.0) / fma(x, (x * x), 1.0)), fma(x, (x + -1.0), 1.0), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = fma(Float64(fma(x, x, -1.0) / fma(x, Float64(x * x), 1.0)), fma(x, Float64(x + -1.0), 1.0), 1.0);
	elseif (t_0 <= 2.0)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(x * x + -1.0), $MachinePrecision] / N[(x * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 (*.f64 #s(literal -2 binary64) x)) < 0.0

   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. +-commutative N/A

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

      2. lower-+.f64 5.3

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

   5. Simplified 5.3%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. flip3-+ N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied egg-rr 20.8%

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

   if 0.0 < (exp.f64 (*.f64 #s(literal -2 binary64) x)) < 2

   1. Initial program 8.4%

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

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)}} - 1 \]
   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

Alternative 3: 79.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ t_1 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, -1\right)}{t\_1}, \mathsf{fma}\left(x, x + -1, 1\right), 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), t\_1, x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))) (t_1 (* x (* x x))))
   (if (<= t_0 0.0)
     (fma (/ (fma x x -1.0) t_1) (fma x (+ x -1.0) 1.0) 1.0)
     (if (<= t_0 2.0)
       (fma (fma (* x x) 0.13333333333333333 -0.3333333333333333) t_1 x)
       -1.0))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double t_1 = x * (x * x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fma((fma(x, x, -1.0) / t_1), fma(x, (x + -1.0), 1.0), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), t_1, x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	t_1 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = fma(Float64(fma(x, x, -1.0) / t_1), fma(x, Float64(x + -1.0), 1.0), 1.0);
	elseif (t_0 <= 2.0)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), t_1, x);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(x * x + -1.0), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(x * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * t$95$1 + x), $MachinePrecision], -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 (*.f64 #s(literal -2 binary64) x)) < 0.0

   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. +-commutative N/A

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

      2. lower-+.f64 5.3

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

   5. Simplified 5.3%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. flip3-+ N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied egg-rr 20.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, -1\right)}{\color{blue}{{x}^{3}}}, \mathsf{fma}\left(x, x + -1, 1\right), 1\right) \]
   9. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 20.8

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

   10. Simplified 20.8%

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

   if 0.0 < (exp.f64 (*.f64 #s(literal -2 binary64) x)) < 2

   1. Initial program 8.4%

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

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)}} - 1 \]
   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

Alternative 4: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-2 \cdot x} \leq 1.001:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.3333333333333333, 2\right), -2\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* -2.0 x)) 1.001)
   x
   (+ -1.0 (/ 2.0 (fma x (fma x (fma x -1.3333333333333333 2.0) -2.0) 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (exp((-2.0 * x)) <= 1.001) {
		tmp = x;
	} else {
		tmp = -1.0 + (2.0 / fma(x, fma(x, fma(x, -1.3333333333333333, 2.0), -2.0), 2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(-2.0 * x)) <= 1.001)
		tmp = x;
	else
		tmp = Float64(-1.0 + Float64(2.0 / fma(x, fma(x, fma(x, -1.3333333333333333, 2.0), -2.0), 2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision], 1.001], x, N[(-1.0 + N[(2.0 / N[(x * N[(x * N[(x * -1.3333333333333333 + 2.0), $MachinePrecision] + -2.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 #s(literal -2 binary64) x)) < 1.0009999999999999

   1. Initial program 36.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 7.1

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

   5. Simplified 7.1%

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

      1. associate--l+ N/A

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

      2. metadata-eval N/A

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

      3. +-rgt-identity 70.8

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

   7. Applied egg-rr 70.8%

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

   if 1.0009999999999999 < (exp.f64 (*.f64 #s(literal -2 binary64) x))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-2 \cdot x} \leq 1.001:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.3333333333333333, 2\right), -2\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, x + -1, 1\right)\\ t_1 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -1.22:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), t\_1, x\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, -1\right)}{\mathsf{fma}\left(t\_1, t\_1, -1\right)} \cdot \mathsf{fma}\left(x, x \cdot x, -1\right), t\_0, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, -1\right)}{t\_1}, t\_0, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma x (+ x -1.0) 1.0)) (t_1 (* x (* x x))))
   (if (<= x -1.22)
     -1.0
     (if (<= x 1.6)
       (fma (fma (* x x) 0.13333333333333333 -0.3333333333333333) t_1 x)
       (if (<= x 5.8e+102)
         (fma
          (* (/ (fma x x -1.0) (fma t_1 t_1 -1.0)) (fma x (* x x) -1.0))
          t_0
          1.0)
         (fma (/ (fma x x -1.0) t_1) t_0 1.0))))))

C

double code(double x, double y) {
	double t_0 = fma(x, (x + -1.0), 1.0);
	double t_1 = x * (x * x);
	double tmp;
	if (x <= -1.22) {
		tmp = -1.0;
	} else if (x <= 1.6) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), t_1, x);
	} else if (x <= 5.8e+102) {
		tmp = fma(((fma(x, x, -1.0) / fma(t_1, t_1, -1.0)) * fma(x, (x * x), -1.0)), t_0, 1.0);
	} else {
		tmp = fma((fma(x, x, -1.0) / t_1), t_0, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(x, Float64(x + -1.0), 1.0)
	t_1 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -1.22)
		tmp = -1.0;
	elseif (x <= 1.6)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), t_1, x);
	elseif (x <= 5.8e+102)
		tmp = fma(Float64(Float64(fma(x, x, -1.0) / fma(t_1, t_1, -1.0)) * fma(x, Float64(x * x), -1.0)), t_0, 1.0);
	else
		tmp = fma(Float64(fma(x, x, -1.0) / t_1), t_0, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.22], -1.0, If[LessEqual[x, 1.6], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * t$95$1 + x), $MachinePrecision], If[LessEqual[x, 5.8e+102], N[(N[(N[(N[(x * x + -1.0), $MachinePrecision] / N[(t$95$1 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision], N[(N[(N[(x * x + -1.0), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.21999999999999997

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -1.21999999999999997 < x < 1.6000000000000001

   1. Initial program 8.4%

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

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

   if 1.6000000000000001 < x < 5.8000000000000005e102

   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. +-commutative N/A

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

      2. lower-+.f64 7.7

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

   5. Simplified 7.7%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. flip3-+ N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied egg-rr 7.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   9. Applied egg-rr 66.9%

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

   if 5.8000000000000005e102 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 4.2

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

   5. Simplified 4.2%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. flip3-+ N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied egg-rr 26.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, -1\right)}{\color{blue}{{x}^{3}}}, \mathsf{fma}\left(x, x + -1, 1\right), 1\right) \]
   9. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 26.8

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

   10. Simplified 26.8%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, -1\right)}{\color{blue}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(x, x + -1, 1\right), 1\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 82.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, x + -1, 1\right)\\ t_1 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -1.22:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), t\_1, x\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, t\_1, -1\right)}{\mathsf{fma}\left(x, x \cdot x, 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)}, t\_0, 1\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot x, -1\right), \frac{\mathsf{fma}\left(x, x, 1 - x\right)}{\mathsf{fma}\left(x, t\_1, 1 + x \cdot x\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, -1\right)}{t\_1}, t\_0, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma x (+ x -1.0) 1.0)) (t_1 (* x (* x x))))
   (if (<= x -1.22)
     -1.0
     (if (<= x 1.6)
       (fma (fma (* x x) 0.13333333333333333 -0.3333333333333333) t_1 x)
       (if (<= x 1.15e+77)
         (fma
          (/ (fma x t_1 -1.0) (* (fma x (* x x) 1.0) (fma x x 1.0)))
          t_0
          1.0)
         (if (<= x 5.8e+102)
           (fma
            (fma x (* x x) -1.0)
            (/ (fma x x (- 1.0 x)) (fma x t_1 (+ 1.0 (* x x))))
            1.0)
           (fma (/ (fma x x -1.0) t_1) t_0 1.0)))))))

C

double code(double x, double y) {
	double t_0 = fma(x, (x + -1.0), 1.0);
	double t_1 = x * (x * x);
	double tmp;
	if (x <= -1.22) {
		tmp = -1.0;
	} else if (x <= 1.6) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), t_1, x);
	} else if (x <= 1.15e+77) {
		tmp = fma((fma(x, t_1, -1.0) / (fma(x, (x * x), 1.0) * fma(x, x, 1.0))), t_0, 1.0);
	} else if (x <= 5.8e+102) {
		tmp = fma(fma(x, (x * x), -1.0), (fma(x, x, (1.0 - x)) / fma(x, t_1, (1.0 + (x * x)))), 1.0);
	} else {
		tmp = fma((fma(x, x, -1.0) / t_1), t_0, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(x, Float64(x + -1.0), 1.0)
	t_1 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -1.22)
		tmp = -1.0;
	elseif (x <= 1.6)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), t_1, x);
	elseif (x <= 1.15e+77)
		tmp = fma(Float64(fma(x, t_1, -1.0) / Float64(fma(x, Float64(x * x), 1.0) * fma(x, x, 1.0))), t_0, 1.0);
	elseif (x <= 5.8e+102)
		tmp = fma(fma(x, Float64(x * x), -1.0), Float64(fma(x, x, Float64(1.0 - x)) / fma(x, t_1, Float64(1.0 + Float64(x * x)))), 1.0);
	else
		tmp = fma(Float64(fma(x, x, -1.0) / t_1), t_0, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.22], -1.0, If[LessEqual[x, 1.6], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * t$95$1 + x), $MachinePrecision], If[LessEqual[x, 1.15e+77], N[(N[(N[(x * t$95$1 + -1.0), $MachinePrecision] / N[(N[(x * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision], If[LessEqual[x, 5.8e+102], N[(N[(x * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(x * x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$1 + N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(x * x + -1.0), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.21999999999999997

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -1.21999999999999997 < x < 1.6000000000000001

   1. Initial program 8.4%

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

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

   if 1.6000000000000001 < x < 1.14999999999999997e77

   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. +-commutative N/A

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

      2. lower-+.f64 8.5

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

   5. Simplified 8.5%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. flip3-+ N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied egg-rr 8.5%

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

      1. lift-*.f64 N/A

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

      2. flip-+ N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. sub-neg N/A

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

      16. lift-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 44.5

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

   9. Applied egg-rr 44.5%

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

   if 1.14999999999999997e77 < x < 5.8000000000000005e102

   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. +-commutative N/A

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

      2. lower-+.f64 5.9

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

   5. Simplified 5.9%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. flip3-+ N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied egg-rr 5.9%

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

   9. Applied egg-rr 100.0%

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

   if 5.8000000000000005e102 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 4.2

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

   5. Simplified 4.2%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. flip3-+ N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied egg-rr 26.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, -1\right)}{\color{blue}{{x}^{3}}}, \mathsf{fma}\left(x, x + -1, 1\right), 1\right) \]
   9. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 26.8

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

   10. Simplified 26.8%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, -1\right)}{\color{blue}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(x, x + -1, 1\right), 1\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -1.22:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), t\_0, x\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot x, -1\right), \frac{\mathsf{fma}\left(x, x, 1 - x\right)}{\mathsf{fma}\left(x, t\_0, 1 + x \cdot x\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, -1\right)}{t\_0}, \mathsf{fma}\left(x, x + -1, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -1.22)
     -1.0
     (if (<= x 1.6)
       (fma (fma (* x x) 0.13333333333333333 -0.3333333333333333) t_0 x)
       (if (<= x 5.8e+102)
         (fma
          (fma x (* x x) -1.0)
          (/ (fma x x (- 1.0 x)) (fma x t_0 (+ 1.0 (* x x))))
          1.0)
         (fma (/ (fma x x -1.0) t_0) (fma x (+ x -1.0) 1.0) 1.0))))))

C

double code(double x, double y) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -1.22) {
		tmp = -1.0;
	} else if (x <= 1.6) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), t_0, x);
	} else if (x <= 5.8e+102) {
		tmp = fma(fma(x, (x * x), -1.0), (fma(x, x, (1.0 - x)) / fma(x, t_0, (1.0 + (x * x)))), 1.0);
	} else {
		tmp = fma((fma(x, x, -1.0) / t_0), fma(x, (x + -1.0), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -1.22)
		tmp = -1.0;
	elseif (x <= 1.6)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), t_0, x);
	elseif (x <= 5.8e+102)
		tmp = fma(fma(x, Float64(x * x), -1.0), Float64(fma(x, x, Float64(1.0 - x)) / fma(x, t_0, Float64(1.0 + Float64(x * x)))), 1.0);
	else
		tmp = fma(Float64(fma(x, x, -1.0) / t_0), fma(x, Float64(x + -1.0), 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.22], -1.0, If[LessEqual[x, 1.6], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * t$95$0 + x), $MachinePrecision], If[LessEqual[x, 5.8e+102], N[(N[(x * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(x * x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0 + N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(x * x + -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(x * N[(x + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.21999999999999997

   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

   if -1.21999999999999997 < x < 1.6000000000000001

   1. Initial program 8.4%

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

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

   if 1.6000000000000001 < x < 5.8000000000000005e102

   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. +-commutative N/A

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

      2. lower-+.f64 7.7

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

   5. Simplified 7.7%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. flip3-+ N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied egg-rr 7.7%

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

   9. Applied egg-rr 37.4%

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

   if 5.8000000000000005e102 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 4.2

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

   5. Simplified 4.2%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. flip-- N/A

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

      5. flip3-+ N/A

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

      6. associate-/r/ N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied egg-rr 26.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, -1\right)}{\color{blue}{{x}^{3}}}, \mathsf{fma}\left(x, x + -1, 1\right), 1\right) \]
   9. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 26.8

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

   10. Simplified 26.8%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x, -1\right)}{\color{blue}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(x, x + -1, 1\right), 1\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* -2.0 x) 0.002) x -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= 0.002) {
		tmp = x;
	} else {
		tmp = -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 (((-2.0d0) * x) <= 0.002d0) then
        tmp = x
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= 0.002) {
		tmp = x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= 0.002:
		tmp = x
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= 0.002)
		tmp = x;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= 0.002)
		tmp = x;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], x, -1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 36.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 7.3

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

   5. Simplified 7.3%

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

      1. associate--l+ N/A

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

      2. metadata-eval N/A

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

      3. +-rgt-identity 70.8

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

   7. Applied egg-rr 70.8%

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

   if 2e-3 < (*.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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

Alternative 9: 29.1% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-154}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.12e-154) -1.0 0.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.12e-154) {
		tmp = -1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.12d-154)) then
        tmp = -1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.12e-154) {
		tmp = -1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.12e-154:
		tmp = -1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.12e-154)
		tmp = -1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.12e-154)
		tmp = -1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.12e-154], -1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.12e-154

   1. Initial program 67.9%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 67.7

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

   5. Simplified 67.7%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 66.7%

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

   if -1.12e-154 < x

   1. Initial program 41.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 4.8%

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

      1. metadata-eval 4.8

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

   7. Applied egg-rr 4.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 27.5% accurate, 123.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 51.0%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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. lower-fma.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 27.8

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

5. Simplified 27.8%

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

6. Taylor expanded in x around inf

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

8. Simplified 25.5%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))