Logistic function from Lakshay Garg

Percentage Accurate: 54.5% → 100.0%

Time: 11.6s

Alternatives: 16

Speedup: 4.7×

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: 54.5% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.024 \lor \neg \left(x \leq 0.0245\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{7}, -0.05396825396825397, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.024) (not (<= x 0.0245)))
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (fma
    (pow x 7.0)
    -0.05396825396825397
    (fma
     (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
     x
     x))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.024) || !(x <= 0.0245)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else {
		tmp = fma(pow(x, 7.0), -0.05396825396825397, fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.024) || !(x <= 0.0245))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	else
		tmp = fma((x ^ 7.0), -0.05396825396825397, 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, -0.024], N[Not[LessEqual[x, 0.0245]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[Power[x, 7.0], $MachinePrecision] * -0.05396825396825397 + N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.024 or 0.024500000000000001 < x

   1. Initial program 100.0%

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

   if -0.024 < x < 0.024500000000000001

   1. Initial program 8.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left({x}^{4}, -0.05396825396825397, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right), x\right)} \]

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left({x}^{7}, \color{blue}{-0.05396825396825397}, \mathsf{fma}\left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, {x}^{3}, x\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left({x}^{7}, -0.05396825396825397, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.024 \lor \neg \left(x \leq 0.0245\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({x}^{7}, -0.05396825396825397, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \leq -0.5:\\ \;\;\;\;\frac{2}{\left(-1.3333333333333333 \cdot x\right) \cdot \left(x \cdot x\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0) -0.5)
   (- (/ 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 (((2.0 / (1.0 + exp((-2.0 * x)))) - 1.0) <= -0.5) {
		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 (Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0) <= -0.5)
		tmp = Float64(Float64(2.0 / Float64(Float64(-1.3333333333333333 * x) * Float64(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[N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.5], N[(N[(2.0 / N[(N[(-1.3333333333333333 * x), $MachinePrecision] * N[(x * x), $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 (-.f64 (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) #s(literal 1 binary64)) < -0.5

   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 99.0

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

   5. Applied rewrites 99.0%

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

   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 99.0%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.0%

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

   if -0.5 < (-.f64 (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) #s(literal 1 binary64))

   1. Initial program 34.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 73.0%

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

Alternative 3: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} - 1 \leq -0.5:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, 2, -2\right), x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0) -0.5)
   (- (/ 2.0 (fma (fma x 2.0 -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 (((2.0 / (1.0 + exp((-2.0 * x)))) - 1.0) <= -0.5) {
		tmp = (2.0 / fma(fma(x, 2.0, -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 (Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0) <= -0.5)
		tmp = Float64(Float64(2.0 / fma(fma(x, 2.0, -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[N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.5], N[(N[(2.0 / N[(N[(x * 2.0 + -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 (-.f64 (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) #s(literal 1 binary64)) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. rgt-mult-inverse N/A

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

      7. distribute-lft-neg-out N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. rgt-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 98.6

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

   8. Applied rewrites 98.6%

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

   if -0.5 < (-.f64 (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) #s(literal 1 binary64))

   1. Initial program 34.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 73.0%

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

Alternative 4: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.024 \lor \neg \left(x \leq 0.0245\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, {x}^{4}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.024) (not (<= x 0.0245)))
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (fma
    (*
     (fma
      -0.05396825396825397
      (pow x 4.0)
      (- (* 0.13333333333333333 (* x x)) 0.3333333333333333))
     (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.024) || !(x <= 0.0245)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else {
		tmp = fma((fma(-0.05396825396825397, pow(x, 4.0), ((0.13333333333333333 * (x * x)) - 0.3333333333333333)) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.024) || !(x <= 0.0245))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	else
		tmp = fma(Float64(fma(-0.05396825396825397, (x ^ 4.0), Float64(Float64(0.13333333333333333 * Float64(x * x)) - 0.3333333333333333)) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.024], N[Not[LessEqual[x, 0.0245]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(-0.05396825396825397 * N[Power[x, 4.0], $MachinePrecision] + N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.024 or 0.024500000000000001 < x

   1. Initial program 100.0%

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

   if -0.024 < x < 0.024500000000000001

   1. Initial program 8.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left({x}^{4}, -0.05396825396825397, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, {x}^{4}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.024 \lor \neg \left(x \leq 0.0245\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, {x}^{4}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.001 \lor \neg \left(x \leq 0.0009\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.001) (not (<= x 0.0009)))
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (* (fma (* x x) -0.3333333333333333 1.0) x)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.001) || !(x <= 0.0009)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else {
		tmp = fma((x * x), -0.3333333333333333, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.001) || !(x <= 0.0009))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	else
		tmp = Float64(fma(Float64(x * x), -0.3333333333333333, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.001], N[Not[LessEqual[x, 0.0009]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e-3 or 8.9999999999999998e-4 < x

   1. Initial program 99.9%

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

   if -1e-3 < x < 8.9999999999999998e-4

   1. Initial program 8.3%

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

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

   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 100.0%

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

   12. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.001 \lor \neg \left(x \leq 0.0009\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (- (/ 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 <= -1.0) {
		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 <= -1.0)
		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, -1.0], 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 < -1

   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 99.0

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

   5. Applied rewrites 99.0%

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

   if -1 < x

   1. Initial program 34.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 73.0%

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

Alternative 7: 75.6% 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 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 99.0%

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

   if -1.1499999999999999 < x

   1. Initial program 34.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 73.0%

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

Alternative 8: 75.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(-1.3333333333333333, x, 2\right) \cdot \left(x \cdot x\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.35)
   (- (/ 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 <= -1.35) {
		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 <= -1.35)
		tmp = Float64(Float64(2.0 / Float64(fma(-1.3333333333333333, x, 2.0) * Float64(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, -1.35], N[(N[(2.0 / N[(N[(-1.3333333333333333 * x + 2.0), $MachinePrecision] * N[(x * x), $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 < -1.3500000000000001

   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 99.0

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

   5. Applied rewrites 99.0%

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

   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 99.0%

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

   if -1.3500000000000001 < x

   1. Initial program 34.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 73.0%

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

Alternative 9: 74.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(N[(2.0 / N[(N[(x * 2.0 + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] * 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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. rgt-mult-inverse N/A

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

      7. distribute-lft-neg-out N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. rgt-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 98.6

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

   8. Applied rewrites 98.6%

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

   if -1 < x

   1. Initial program 34.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 73.0%

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

   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 72.1%

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

   12. Applied rewrites 72.1%

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

Alternative 10: 74.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2 \cdot x, x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.2)
   (- (/ 2.0 (fma (* 2.0 x) x 2.0)) 1.0)
   (* (fma (* x x) -0.3333333333333333 1.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = (2.0 / fma((2.0 * x), x, 2.0)) - 1.0;
	} else {
		tmp = fma((x * x), -0.3333333333333333, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.2)
		tmp = Float64(Float64(2.0 / fma(Float64(2.0 * x), x, 2.0)) - 1.0);
	else
		tmp = Float64(fma(Float64(x * x), -0.3333333333333333, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.2], N[(N[(2.0 / N[(N[(2.0 * x), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. rgt-mult-inverse N/A

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

      7. distribute-lft-neg-out N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. rgt-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 98.6

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 98.6%

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

   if -1.19999999999999996 < x

   1. Initial program 34.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 73.0%

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

   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 72.1%

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

   12. Applied rewrites 72.1%

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

Alternative 11: 74.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.4) {
		tmp = (2.0 / ((2.0 * x) * x)) - 1.0;
	} else {
		tmp = fma((x * x), -0.3333333333333333, 1.0) * 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 = Float64(fma(Float64(x * x), -0.3333333333333333, 1.0) * 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[(N[(x * x), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] * 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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. rgt-mult-inverse N/A

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

      7. distribute-lft-neg-out N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. rgt-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 98.6

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 98.6%

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

   if -1.3999999999999999 < x

   1. Initial program 34.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 73.0%

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

   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 72.1%

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

   12. Applied rewrites 72.1%

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

Alternative 12: 74.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(-2, x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.35)
   (- (/ 2.0 (fma -2.0 x 2.0)) 1.0)
   (* (fma (* x x) -0.3333333333333333 1.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.35) {
		tmp = (2.0 / fma(-2.0, x, 2.0)) - 1.0;
	} else {
		tmp = fma((x * x), -0.3333333333333333, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.35)
		tmp = Float64(Float64(2.0 / fma(-2.0, x, 2.0)) - 1.0);
	else
		tmp = Float64(fma(Float64(x * x), -0.3333333333333333, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.35], N[(N[(2.0 / N[(-2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3500000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 96.9

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

   5. Applied rewrites 96.9%

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

   if -1.3500000000000001 < x

   1. Initial program 34.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 73.0%

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

   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 72.1%

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

   12. Applied rewrites 72.1%

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

Alternative 13: 74.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\frac{2}{x \cdot -2} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.5)
   (- (/ 2.0 (* x -2.0)) 1.0)
   (* (fma (* x x) -0.3333333333333333 1.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.5) {
		tmp = (2.0 / (x * -2.0)) - 1.0;
	} else {
		tmp = fma((x * x), -0.3333333333333333, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.5)
		tmp = Float64(Float64(2.0 / Float64(x * -2.0)) - 1.0);
	else
		tmp = Float64(fma(Float64(x * x), -0.3333333333333333, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.5], N[(N[(2.0 / N[(x * -2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 96.9%

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

   if -1.5 < x

   1. Initial program 34.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 73.0%

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

   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 72.1%

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

   12. Applied rewrites 72.1%

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

Alternative 14: 49.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fma (* x x) -0.3333333333333333 1.0) x))

C

double code(double x) {
	return fma((x * x), -0.3333333333333333, 1.0) * x;
}

Julia

function code(x)
	return Float64(fma(Float64(x * x), -0.3333333333333333, 1.0) * x)
end

Wolfram

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

Derivation

1. Initial program 51.3%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
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 55.1

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

5. Applied rewrites 55.1%

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

7. Applied rewrites 55.1%

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

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 53.7%

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

12. Applied rewrites 53.7%

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

Alternative 15: 6.6% 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 51.3%

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

3. Taylor expanded in x around 0

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

   1. lower-+.f64 7.0

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

5. Applied rewrites 7.0%

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

Alternative 16: 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 51.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.3%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))