Logistic function from Lakshay Garg

Percentage Accurate: 53.9% → 100.0%

Time: 8.9s

Alternatives: 19

Speedup: 5.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{x}\right)}^{-2}\\ t_1 := \log 2 - \mathsf{log1p}\left(t\_0\right)\\ t_2 := \frac{2}{t\_0 + 1}\\ t_3 := t\_2 + 1\\ \mathbf{if}\;x \leq -0.0076:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \leq 0.0074:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{\frac{{t\_2}^{4}}{t\_3}}{\mathsf{expm1}\left(t\_1 \cdot 2\right)}}, \sqrt{\mathsf{expm1}\left(t\_1\right)}, \frac{-1}{t\_3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (exp x) -2.0))
        (t_1 (- (log 2.0) (log1p t_0)))
        (t_2 (/ 2.0 (+ t_0 1.0)))
        (t_3 (+ t_2 1.0)))
   (if (<= x -0.0076)
     (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
     (if (<= x 0.0074)
       (fma
        (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
        x
        x)
       (fma
        (sqrt (/ (/ (pow t_2 4.0) t_3) (expm1 (* t_1 2.0))))
        (sqrt (expm1 t_1))
        (/ -1.0 t_3))))))

C

double code(double x) {
	double t_0 = pow(exp(x), -2.0);
	double t_1 = log(2.0) - log1p(t_0);
	double t_2 = 2.0 / (t_0 + 1.0);
	double t_3 = t_2 + 1.0;
	double tmp;
	if (x <= -0.0076) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else if (x <= 0.0074) {
		tmp = fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x);
	} else {
		tmp = fma(sqrt(((pow(t_2, 4.0) / t_3) / expm1((t_1 * 2.0)))), sqrt(expm1(t_1)), (-1.0 / t_3));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(x) ^ -2.0
	t_1 = Float64(log(2.0) - log1p(t_0))
	t_2 = Float64(2.0 / Float64(t_0 + 1.0))
	t_3 = Float64(t_2 + 1.0)
	tmp = 0.0
	if (x <= -0.0076)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	elseif (x <= 0.0074)
		tmp = fma(Float64(Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333) * Float64(x * x)), x, x);
	else
		tmp = fma(sqrt(Float64(Float64((t_2 ^ 4.0) / t_3) / expm1(Float64(t_1 * 2.0)))), sqrt(expm1(t_1)), Float64(-1.0 / t_3));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + 1.0), $MachinePrecision]}, If[LessEqual[x, -0.0076], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 0.0074], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[Sqrt[N[(N[(N[Power[t$95$2, 4.0], $MachinePrecision] / t$95$3), $MachinePrecision] / N[(Exp[N[(t$95$1 * 2.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Exp[t$95$1] - 1), $MachinePrecision]], $MachinePrecision] + N[(-1.0 / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00759999999999999998

   1. Initial program 100.0%

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

   if -0.00759999999999999998 < x < 0.0074000000000000003

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   if 0.0074000000000000003 < x

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.9

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

      4. lift-exp.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   6. Applied rewrites 100.0%

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

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{\frac{{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1}\right)}^{4}}{\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + 1}}{\mathsf{expm1}\left(\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right) \cdot 2\right)}}, \sqrt{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)}, \frac{-1}{\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + 1}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{x}\right)}^{-2}\\ t_1 := t\_0 + 1\\ t_2 := \frac{2}{t\_1} + 1\\ \mathbf{if}\;x \leq -0.0076:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \leq 0.008:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{1 + t\_0}, \sqrt{\frac{\frac{4}{{t\_1}^{2}}}{{t\_2}^{2}}}, \frac{-1}{t\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (exp x) -2.0)) (t_1 (+ t_0 1.0)) (t_2 (+ (/ 2.0 t_1) 1.0)))
   (if (<= x -0.0076)
     (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
     (if (<= x 0.008)
       (fma
        (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
        x
        x)
       (fma
        (/ 2.0 (+ 1.0 t_0))
        (sqrt (/ (/ 4.0 (pow t_1 2.0)) (pow t_2 2.0)))
        (/ -1.0 t_2))))))

C

double code(double x) {
	double t_0 = pow(exp(x), -2.0);
	double t_1 = t_0 + 1.0;
	double t_2 = (2.0 / t_1) + 1.0;
	double tmp;
	if (x <= -0.0076) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else if (x <= 0.008) {
		tmp = fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x);
	} else {
		tmp = fma((2.0 / (1.0 + t_0)), sqrt(((4.0 / pow(t_1, 2.0)) / pow(t_2, 2.0))), (-1.0 / t_2));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(x) ^ -2.0
	t_1 = Float64(t_0 + 1.0)
	t_2 = Float64(Float64(2.0 / t_1) + 1.0)
	tmp = 0.0
	if (x <= -0.0076)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	elseif (x <= 0.008)
		tmp = fma(Float64(Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333) * Float64(x * x)), x, x);
	else
		tmp = fma(Float64(2.0 / Float64(1.0 + t_0)), sqrt(Float64(Float64(4.0 / (t_1 ^ 2.0)) / (t_2 ^ 2.0))), Float64(-1.0 / t_2));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -0.0076], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 0.008], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(2.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(4.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00759999999999999998

   1. Initial program 100.0%

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

   if -0.00759999999999999998 < x < 0.0080000000000000002

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   if 0.0080000000000000002 < x

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.9

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

      4. lift-exp.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   6. Applied rewrites 100.0%

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

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{4}{{\left({\left(e^{x}\right)}^{-2} + 1\right)}^{2}}}, \sqrt{\frac{\frac{4}{{\left({\left(e^{x}\right)}^{-2} + 1\right)}^{2}}}{{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + 1\right)}^{2}}}, \frac{-1}{\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + 1}\right)} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f64 N/A

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

      6. sqrt-pow1 N/A

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

      7. metadata-eval N/A

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

      8. unpow1 N/A

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

      9. lift-/.f64 100.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   10. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{1 + {\left(e^{x}\right)}^{-2}}}, \sqrt{\frac{\frac{4}{{\left({\left(e^{x}\right)}^{-2} + 1\right)}^{2}}}{{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + 1\right)}^{2}}}, \frac{-1}{\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + 1}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{x}\right)}^{-2} + 1\\ t_1 := \frac{2}{t\_0} + 1\\ \mathbf{if}\;x \leq -0.0076:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \leq 0.008:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4}{{t\_0}^{2}}}{t\_1} + \frac{-1}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (pow (exp x) -2.0) 1.0)) (t_1 (+ (/ 2.0 t_0) 1.0)))
   (if (<= x -0.0076)
     (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
     (if (<= x 0.008)
       (fma
        (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
        x
        x)
       (+ (/ (/ 4.0 (pow t_0 2.0)) t_1) (/ -1.0 t_1))))))

C

double code(double x) {
	double t_0 = pow(exp(x), -2.0) + 1.0;
	double t_1 = (2.0 / t_0) + 1.0;
	double tmp;
	if (x <= -0.0076) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else if (x <= 0.008) {
		tmp = fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x);
	} else {
		tmp = ((4.0 / pow(t_0, 2.0)) / t_1) + (-1.0 / t_1);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64((exp(x) ^ -2.0) + 1.0)
	t_1 = Float64(Float64(2.0 / t_0) + 1.0)
	tmp = 0.0
	if (x <= -0.0076)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	elseif (x <= 0.008)
		tmp = fma(Float64(Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333) * Float64(x * x)), x, x);
	else
		tmp = Float64(Float64(Float64(4.0 / (t_0 ^ 2.0)) / t_1) + Float64(-1.0 / t_1));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -0.0076], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 0.008], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(N[(4.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00759999999999999998

   1. Initial program 100.0%

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

   if -0.00759999999999999998 < x < 0.0080000000000000002

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   if 0.0080000000000000002 < x

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.9

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

      4. lift-exp.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   6. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. distribute-lft-neg-in N/A

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

      6. sqr-pow N/A

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

      7. lift-pow.f64 N/A

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

   8. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\frac{4}{{\left({\left(e^{x}\right)}^{-2} + 1\right)}^{2}}}{\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + 1} + \frac{-1}{\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + 1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ t_1 := \frac{2}{1 + {\left(e^{x}\right)}^{-2}}\\ \mathbf{if}\;x \leq -0.0076:\\ \;\;\;\;t\_0 - 1\\ \mathbf{elif}\;x \leq 0.0075:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_1}^{2}}{t\_1 + 1} - {\left(t\_0 + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
        (t_1 (/ 2.0 (+ 1.0 (pow (exp x) -2.0)))))
   (if (<= x -0.0076)
     (- t_0 1.0)
     (if (<= x 0.0075)
       (fma
        (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
        x
        x)
       (- (/ (pow t_1 2.0) (+ t_1 1.0)) (pow (+ t_0 1.0) -1.0))))))

C

double code(double x) {
	double t_0 = 2.0 / (1.0 + exp((-2.0 * x)));
	double t_1 = 2.0 / (1.0 + pow(exp(x), -2.0));
	double tmp;
	if (x <= -0.0076) {
		tmp = t_0 - 1.0;
	} else if (x <= 0.0075) {
		tmp = fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x);
	} else {
		tmp = (pow(t_1, 2.0) / (t_1 + 1.0)) - pow((t_0 + 1.0), -1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x))))
	t_1 = Float64(2.0 / Float64(1.0 + (exp(x) ^ -2.0)))
	tmp = 0.0
	if (x <= -0.0076)
		tmp = Float64(t_0 - 1.0);
	elseif (x <= 0.0075)
		tmp = fma(Float64(Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333) * Float64(x * x)), x, x);
	else
		tmp = Float64(Float64((t_1 ^ 2.0) / Float64(t_1 + 1.0)) - (Float64(t_0 + 1.0) ^ -1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(1.0 + N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0076], N[(t$95$0 - 1.0), $MachinePrecision], If[LessEqual[x, 0.0075], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] - N[Power[N[(t$95$0 + 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00759999999999999998

   1. Initial program 100.0%

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

   if -0.00759999999999999998 < x < 0.0074999999999999997

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   if 0.0074999999999999997 < x

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.9

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

      4. lift-exp.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   6. Applied rewrites 100.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. pow-exp N/A

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

      4. *-commutative N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

Alternative 5: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0076:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \leq 0.0074:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(e^{x}\right)}^{-2} + 1} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0076)
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (if (<= x 0.0074)
     (fma
      (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
      x
      x)
     (- (/ 2.0 (+ (pow (exp x) -2.0) 1.0)) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0076) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else if (x <= 0.0074) {
		tmp = fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x);
	} else {
		tmp = (2.0 / (pow(exp(x), -2.0) + 1.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0076)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	elseif (x <= 0.0074)
		tmp = fma(Float64(Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333) * Float64(x * x)), x, x);
	else
		tmp = Float64(Float64(2.0 / Float64((exp(x) ^ -2.0) + 1.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.0076], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 0.0074], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(2.0 / N[(N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00759999999999999998

   1. Initial program 100.0%

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

   if -0.00759999999999999998 < x < 0.0074000000000000003

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   if 0.0074000000000000003 < x

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.9

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

      4. lift-exp.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 6: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0076 \lor \neg \left(x \leq 0.0074\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.0076) (not (<= x 0.0074)))
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (fma
    (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.0076) || !(x <= 0.0074)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else {
		tmp = fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.0076) || !(x <= 0.0074))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	else
		tmp = fma(Float64(Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.0076], N[Not[LessEqual[x, 0.0074]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00759999999999999998 or 0.0074000000000000003 < x

   1. Initial program 100.0%

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

   if -0.00759999999999999998 < x < 0.0074000000000000003

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0076 \lor \neg \left(x \leq 0.0074\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.98:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right) \cdot x - 2, x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.98)
   (- (/ 2.0 (fma (- (* (fma -1.3333333333333333 x 2.0) x) 2.0) x 2.0)) 1.0)
   (fma
    (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -0.98) {
		tmp = (2.0 / fma(((fma(-1.3333333333333333, x, 2.0) * x) - 2.0), x, 2.0)) - 1.0;
	} else {
		tmp = fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.98)
		tmp = Float64(Float64(2.0 / fma(Float64(Float64(fma(-1.3333333333333333, x, 2.0) * x) - 2.0), x, 2.0)) - 1.0);
	else
		tmp = fma(Float64(Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.98], N[(N[(2.0 / N[(N[(N[(N[(-1.3333333333333333 * x + 2.0), $MachinePrecision] * x), $MachinePrecision] - 2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.97999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if -0.97999999999999998 < x

   1. Initial program 41.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.5%

      \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right) \cdot x - 2\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (- (/ 2.0 (* (- (* (fma -1.3333333333333333 x 2.0) x) 2.0) x)) 1.0)
   (fma
    (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = (2.0 / (((fma(-1.3333333333333333, x, 2.0) * x) - 2.0) * x)) - 1.0;
	} else {
		tmp = fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(fma(-1.3333333333333333, x, 2.0) * x) - 2.0) * x)) - 1.0);
	else
		tmp = fma(Float64(Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.15], N[(N[(2.0 / N[(N[(N[(N[(-1.3333333333333333 * x + 2.0), $MachinePrecision] * x), $MachinePrecision] - 2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.0%

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

   if -1.1499999999999999 < x

   1. Initial program 41.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.5%

      \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right) \cdot x\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (- (/ 2.0 (* (* (fma -1.3333333333333333 x 2.0) x) x)) 1.0)
   (fma
    (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = (2.0 / ((fma(-1.3333333333333333, x, 2.0) * x) * x)) - 1.0;
	} else {
		tmp = fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(Float64(2.0 / Float64(Float64(fma(-1.3333333333333333, x, 2.0) * x) * x)) - 1.0);
	else
		tmp = fma(Float64(Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[(N[(2.0 / N[(N[(N[(-1.3333333333333333 * x + 2.0), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.30000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 98.0%

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

   if -1.30000000000000004 < x

   1. Initial program 41.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.5%

      \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 75.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{2}{\left(\left(-1.3333333333333333 \cdot x\right) \cdot x\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (- (/ 2.0 (* (* (* -1.3333333333333333 x) x) x)) 1.0)
   (fma
    (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = (2.0 / (((-1.3333333333333333 * x) * x) * x)) - 1.0;
	} else {
		tmp = fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(-1.3333333333333333 * x) * x) * x)) - 1.0);
	else
		tmp = fma(Float64(Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(N[(2.0 / N[(N[(N[(-1.3333333333333333 * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 98.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 98.0%

       \[\leadsto \frac{2}{\left(\left(-1.3333333333333333 \cdot x\right) \cdot x\right) \cdot x} - 1 \]

   if -1.55000000000000004 < x

   1. Initial program 41.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.5%

      \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 75.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, -2\right), x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (- (/ 2.0 (fma (fma 2.0 x -2.0) x 2.0)) 1.0)
   (fma
    (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = (2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0;
	} else {
		tmp = fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15)
		tmp = Float64(Float64(2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0);
	else
		tmp = fma(Float64(Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.15], N[(N[(2.0 / N[(N[(2.0 * x + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1499999999999999

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 100.0

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

      4. lift-exp.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.9%

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

   if -1.1499999999999999 < x

   1. Initial program 41.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.5%

      \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, -2\right), x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, -2\right), x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (- (/ 2.0 (fma (fma 2.0 x -2.0) x 2.0)) 1.0)
   (*
    (fma (- (* 0.13333333333333333 (* x x)) 0.3333333333333333) (* x x) 1.0)
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = (2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0;
	} else {
		tmp = fma(((0.13333333333333333 * (x * x)) - 0.3333333333333333), (x * x), 1.0) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15)
		tmp = Float64(Float64(2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0);
	else
		tmp = Float64(fma(Float64(Float64(0.13333333333333333 * Float64(x * x)) - 0.3333333333333333), Float64(x * x), 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.15], N[(N[(2.0 / N[(N[(2.0 * x + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1499999999999999

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 100.0

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

      4. lift-exp.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.9%

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

   if -1.1499999999999999 < x

   1. Initial program 41.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.5%

      \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 65.5%

      \[\leadsto \mathsf{fma}\left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, -2\right), x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 100.0

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

      4. lift-exp.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.9%

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

   if -1 < x

   1. Initial program 41.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.3%

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

4. Final simplification 72.8%

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

Alternative 14: 74.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2 \cdot x, x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (- (/ 2.0 (fma (* 2.0 x) x 2.0)) 1.0)
   (fma (* -0.3333333333333333 (* x x)) x x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = (2.0 / fma((2.0 * x), x, 2.0)) - 1.0;
	} else {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = Float64(Float64(2.0 / fma(Float64(2.0 * x), x, 2.0)) - 1.0);
	else
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[(N[(2.0 / N[(N[(2.0 * x), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 100.0

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

      4. lift-exp.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 97.9%

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

   if -1.25 < x

   1. Initial program 41.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.3%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2 \cdot x, x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 74.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{2}{\left(2 \cdot x\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.4)
   (- (/ 2.0 (* (* 2.0 x) x)) 1.0)
   (fma (* -0.3333333333333333 (* x x)) x x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.4) {
		tmp = (2.0 / ((2.0 * x) * x)) - 1.0;
	} else {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(Float64(2.0 / Float64(Float64(2.0 * x) * x)) - 1.0);
	else
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.4], N[(N[(2.0 / N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 100.0

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

      4. lift-exp.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 97.9%

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

   if -1.3999999999999999 < x

   1. Initial program 41.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.3%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{2}{\left(2 \cdot x\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 74.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\frac{-1}{x - 1} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (- (/ -1.0 (- x 1.0)) 1.0)
   (fma (* -0.3333333333333333 (* x x)) x x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = (-1.0 / (x - 1.0)) - 1.0;
	} else {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(Float64(-1.0 / Float64(x - 1.0)) - 1.0);
	else
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[(N[(-1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.30000000000000004

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 100.0

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

      4. lift-exp.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 5.1

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

   7. Applied rewrites 5.1%

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

   9. Applied rewrites 4.7%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 97.5%

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

   if -1.30000000000000004 < x

   1. Initial program 41.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

   5. Applied rewrites 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 65.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 64.3%

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

Alternative 17: 50.4% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (* -0.3333333333333333 (* x x)) x x))

C

double code(double x) {
	return fma((-0.3333333333333333 * (x * x)), x, x);
}

Julia

function code(x)
	return fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x)
end

Wolfram

code[x_] := N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 56.2%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. pow-plus N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 49.8

       \[\leadsto \mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \color{blue}{\left(x \cdot x\right)} - 0.3333333333333333, x\right) \]

5. Applied rewrites 49.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 49.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 48.2%

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

Alternative 18: 6.5% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \left(1 + x\right) - 1 \end{array} \]

FPCore

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

C

double code(double x) {
	return (1.0 + x) - 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return (1.0 + x) - 1.0

Julia

function code(x)
	return Float64(Float64(1.0 + x) - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + x) - 1.0;
end

Wolfram

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

Derivation

1. Initial program 56.2%

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

3. Taylor expanded in x around 0

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

   1. lower-+.f64 6.6

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

5. Applied rewrites 6.6%

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

6. Final simplification 6.6%

   \[\leadsto \left(1 + x\right) - 1 \]
7. Add Preprocessing

Alternative 19: 4.3% accurate, 30.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 - 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.2%

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

6. Final simplification 4.2%

   \[\leadsto 1 - 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024338 
(FPCore (x)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))