Logistic function from Lakshay Garg

Percentage Accurate: 55.1% → 100.0%

Time: 7.5s

Alternatives: 11

Speedup: 5.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0082:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \leq 0.0072:\\ \;\;\;\;\mathsf{fma}\left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(-2 \cdot x\right) \cdot 2}{\mathsf{expm1}\left(-4 \cdot x\right)} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0082)
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (if (<= x 0.0072)
     (*
      (fma (- (* 0.13333333333333333 (* x x)) 0.3333333333333333) (* x x) 1.0)
      x)
     (- (/ (* (expm1 (* -2.0 x)) 2.0) (expm1 (* -4.0 x))) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0082) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else if (x <= 0.0072) {
		tmp = fma(((0.13333333333333333 * (x * x)) - 0.3333333333333333), (x * x), 1.0) * x;
	} else {
		tmp = ((expm1((-2.0 * x)) * 2.0) / expm1((-4.0 * x))) - 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0082)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	elseif (x <= 0.0072)
		tmp = Float64(fma(Float64(Float64(0.13333333333333333 * Float64(x * x)) - 0.3333333333333333), Float64(x * x), 1.0) * x);
	else
		tmp = Float64(Float64(Float64(expm1(Float64(-2.0 * x)) * 2.0) / expm1(Float64(-4.0 * x))) - 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.0082], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 0.0072], N[(N[(N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision] * 2.0), $MachinePrecision] / N[(Exp[N[(-4.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00820000000000000069

   1. Initial program 99.9%

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

   if -0.00820000000000000069 < x < 0.0071999999999999998

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0

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

      1. +-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. Step-by-step derivation

   9. Applied rewrites 100.0%

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

   if 0.0071999999999999998 < x

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{2}{\mathsf{expm1}\left(-4 \cdot x\right)} \cdot \mathsf{expm1}\left(x \cdot -2\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\mathsf{expm1}\left(-4 \cdot x\right)} \cdot \mathsf{expm1}\left(x \cdot -2\right)} - 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{\mathsf{expm1}\left(-4 \cdot x\right)}} \cdot \mathsf{expm1}\left(x \cdot -2\right) - 1 \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \mathsf{expm1}\left(x \cdot -2\right)}{\mathsf{expm1}\left(-4 \cdot x\right)}} - 1 \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot \mathsf{expm1}\left(x \cdot -2\right)}{\mathsf{expm1}\left(-4 \cdot x\right)}} - 1 \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(x \cdot -2\right) \cdot 2}}{\mathsf{expm1}\left(-4 \cdot x\right)} - 1 \]

      6. lower-*.f64 100.0

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(x \cdot -2\right) \cdot 2}}{\mathsf{expm1}\left(-4 \cdot x\right)} - 1 \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{x \cdot -2}\right) \cdot 2}{\mathsf{expm1}\left(-4 \cdot x\right)} - 1 \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{-2 \cdot x}\right) \cdot 2}{\mathsf{expm1}\left(-4 \cdot x\right)} - 1 \]

      9. lower-*.f64 100.0

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{-2 \cdot x}\right) \cdot 2}{\mathsf{expm1}\left(-4 \cdot x\right)} - 1 \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(-2 \cdot x\right) \cdot 2}{\mathsf{expm1}\left(-4 \cdot x\right)}} - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0082 \lor \neg \left(x \leq 0.0072\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.0082) (not (<= x 0.0072)))
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (*
    (fma (- (* 0.13333333333333333 (* x x)) 0.3333333333333333) (* x x) 1.0)
    x)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.0082) || !(x <= 0.0072)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else {
		tmp = fma(((0.13333333333333333 * (x * x)) - 0.3333333333333333), (x * x), 1.0) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.0082) || !(x <= 0.0072))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	else
		tmp = Float64(fma(Float64(Float64(0.13333333333333333 * Float64(x * x)) - 0.3333333333333333), Float64(x * x), 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.0082], N[Not[LessEqual[x, 0.0072]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00820000000000000069 or 0.0071999999999999998 < x

   1. Initial program 99.9%

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

   if -0.00820000000000000069 < x < 0.0071999999999999998

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0

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

      1. +-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. Step-by-step derivation

   9. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0082 \lor \neg \left(x \leq 0.0072\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right) \cdot x - 2\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.2)
   (- (/ 2.0 (* (- (* (fma -1.3333333333333333 x 2.0) x) 2.0) x)) 1.0)
   (*
    (fma (- (* 0.13333333333333333 (* x x)) 0.3333333333333333) (* x x) 1.0)
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = (2.0 / (((fma(-1.3333333333333333, x, 2.0) * x) - 2.0) * x)) - 1.0;
	} else {
		tmp = fma(((0.13333333333333333 * (x * x)) - 0.3333333333333333), (x * x), 1.0) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.2)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(fma(-1.3333333333333333, x, 2.0) * x) - 2.0) * x)) - 1.0);
	else
		tmp = Float64(fma(Float64(Float64(0.13333333333333333 * Float64(x * x)) - 0.3333333333333333), Float64(x * x), 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.2], 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[(0.13333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)}} - 1 \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   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 99.3%

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

   if -1.19999999999999996 < x

   1. Initial program 40.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 66.7%

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

Alternative 4: 74.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\frac{2}{\left(-1.3333333333333333 \cdot x\right) \cdot \left(x \cdot \left|x\right|\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.5)
   (- (/ 2.0 (* (* -1.3333333333333333 x) (* x (fabs x)))) 1.0)
   (*
    (fma (- (* 0.13333333333333333 (* x x)) 0.3333333333333333) (* x x) 1.0)
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.5) {
		tmp = (2.0 / ((-1.3333333333333333 * x) * (x * fabs(x)))) - 1.0;
	} else {
		tmp = fma(((0.13333333333333333 * (x * x)) - 0.3333333333333333), (x * x), 1.0) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.5)
		tmp = Float64(Float64(2.0 / Float64(Float64(-1.3333333333333333 * x) * Float64(x * abs(x)))) - 1.0);
	else
		tmp = Float64(fma(Float64(Float64(0.13333333333333333 * Float64(x * x)) - 0.3333333333333333), Float64(x * x), 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.5], N[(N[(2.0 / N[(N[(-1.3333333333333333 * x), $MachinePrecision] * N[(x * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5

   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 99.3

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

   5. Applied rewrites 99.3%

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

   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 99.3%

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

   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 99.3%

       \[\leadsto \frac{2}{\left(-1.3333333333333333 \cdot x\right) \cdot \left(x \cdot x\right)} - 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 99.4%

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

   if -1.5 < x

   1. Initial program 40.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 66.7%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\frac{2}{\left(-1.3333333333333333 \cdot x\right) \cdot \left(x \cdot \left|x\right|\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.30000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.3

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

   5. Applied rewrites 99.3%

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

   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 99.3%

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

   if -1.30000000000000004 < x

   1. Initial program 40.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 66.7%

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

Alternative 6: 74.5% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.3

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

   5. Applied rewrites 99.3%

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

   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 99.3%

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

   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 99.3%

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

   if -1.55000000000000004 < x

   1. Initial program 40.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 66.7%

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

Alternative 7: 74.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, 2, -2\right), x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.2)
   (- (/ 2.0 (fma (fma x 2.0 -2.0) x 2.0)) 1.0)
   (*
    (fma (- (* 0.13333333333333333 (* x x)) 0.3333333333333333) (* x x) 1.0)
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = (2.0 / fma(fma(x, 2.0, -2.0), x, 2.0)) - 1.0;
	} else {
		tmp = fma(((0.13333333333333333 * (x * x)) - 0.3333333333333333), (x * x), 1.0) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.2)
		tmp = Float64(Float64(2.0 / fma(fma(x, 2.0, -2.0), x, 2.0)) - 1.0);
	else
		tmp = Float64(fma(Float64(Float64(0.13333333333333333 * Float64(x * x)) - 0.3333333333333333), Float64(x * x), 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.2], N[(N[(2.0 / N[(N[(x * 2.0 + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.19999999999999996

   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 98.1

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

   5. Applied rewrites 98.1%

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

   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 98.7

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

   8. Applied rewrites 98.7%

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

   if -1.19999999999999996 < x

   1. Initial program 40.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 66.7%

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

Alternative 8: 74.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.43:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, 2, -2\right), x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.43) (- (/ 2.0 (fma (fma x 2.0 -2.0) x 2.0)) 1.0) (* 1.0 x)))

C

double code(double x) {
	double tmp;
	if (x <= -0.43) {
		tmp = (2.0 / fma(fma(x, 2.0, -2.0), x, 2.0)) - 1.0;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.43)
		tmp = Float64(Float64(2.0 / fma(fma(x, 2.0, -2.0), x, 2.0)) - 1.0);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.43], N[(N[(2.0 / N[(N[(x * 2.0 + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.429999999999999993

   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 98.1

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

   5. Applied rewrites 98.1%

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

   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 98.7

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

   8. Applied rewrites 98.7%

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

   if -0.429999999999999993 < x

   1. Initial program 40.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 66.7%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 66.4%

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

Alternative 9: 74.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\frac{-1}{-1 + x} - 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.35) (- (/ -1.0 (+ -1.0 x)) 1.0) (* 1.0 x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.35) {
		tmp = (-1.0 / (-1.0 + x)) - 1.0;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.35d0)) then
        tmp = ((-1.0d0) / ((-1.0d0) + x)) - 1.0d0
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.35) {
		tmp = (-1.0 / (-1.0 + x)) - 1.0;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.35:
		tmp = (-1.0 / (-1.0 + x)) - 1.0
	else:
		tmp = 1.0 * x
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.35)
		tmp = (-1.0 / (-1.0 + x)) - 1.0;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if 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

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lft-mult-inverse N/A

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

      3. fp-cancel-sign-sub N/A

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

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

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

      5. lft-mult-inverse N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 5.3

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

   8. Applied rewrites 5.3%

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

   10. Applied rewrites 4.9%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.1%

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

   if -1.3500000000000001 < x

   1. Initial program 40.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 66.7%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 66.4%

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

Alternative 10: 51.3% accurate, 20.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
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 50.1

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

5. Applied rewrites 50.1%

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

7. Applied rewrites 50.1%

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

9. Applied rewrites 50.1%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 50.4%

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

Alternative 11: 4.2% 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 55.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.2%

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

Reproduce

herbie shell --seed 2024340 
(FPCore (x)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))