Logistic function from Lakshay Garg

Percentage Accurate: 54.6% → 99.0%

Time: 11.1s

Alternatives: 17

Speedup: 3.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;-2 \cdot x \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{{t\_1}^{-2}}{1 + \frac{-2}{-1 - t\_0}}, \frac{1}{\frac{-2}{t\_1} + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(-\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))) (t_1 (+ 1.0 t_0)))
   (if (<= (* -2.0 x) -0.01)
     (fma
      4.0
      (/ (pow t_1 -2.0) (+ 1.0 (/ -2.0 (- -1.0 t_0))))
      (/ 1.0 (+ (/ -2.0 t_1) -1.0)))
     (expm1 (- (- x))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double t_1 = 1.0 + t_0;
	double tmp;
	if ((-2.0 * x) <= -0.01) {
		tmp = fma(4.0, (pow(t_1, -2.0) / (1.0 + (-2.0 / (-1.0 - t_0)))), (1.0 / ((-2.0 / t_1) + -1.0)));
	} else {
		tmp = expm1(-(-x));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	t_1 = Float64(1.0 + t_0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.01)
		tmp = fma(4.0, Float64((t_1 ^ -2.0) / Float64(1.0 + Float64(-2.0 / Float64(-1.0 - t_0)))), Float64(1.0 / Float64(Float64(-2.0 / t_1) + -1.0)));
	else
		tmp = expm1(Float64(-Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.01], N[(4.0 * N[(N[Power[t$95$1, -2.0], $MachinePrecision] / N[(1.0 + N[(-2.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[(-2.0 / t$95$1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Exp[(-(-x))] - 1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.0100000000000000002

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

   4. Applied rewrites 99.9%

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

   if -0.0100000000000000002 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 34.3%

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

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-to-exp N/A

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

      7. lower-expm1.f64 N/A

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      13. metadata-eval 34.3

          \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot \color{blue}{-1}\right) \]

   4. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.7

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

   7. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{neg}\left(\left(-x\right)\right)}\right) \]

      4. lower-neg.f64 99.7

         \[\leadsto \mathsf{expm1}\left(\color{blue}{-\left(-x\right)}\right) \]

   9. Applied rewrites 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{{\left(1 + e^{-2 \cdot x}\right)}^{-2}}{1 + \frac{-2}{-1 - e^{-2 \cdot x}}}, \frac{1}{\frac{-2}{1 + e^{-2 \cdot x}} + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(-\left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.01:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(-\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.01)
   (expm1 (- (log 2.0) (log1p (exp (* -2.0 x)))))
   (expm1 (- (- x)))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.01) {
		tmp = expm1((log(2.0) - log1p(exp((-2.0 * x)))));
	} else {
		tmp = expm1(-(-x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.01) {
		tmp = Math.expm1((Math.log(2.0) - Math.log1p(Math.exp((-2.0 * x)))));
	} else {
		tmp = Math.expm1(-(-x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.01:
		tmp = math.expm1((math.log(2.0) - math.log1p(math.exp((-2.0 * x)))))
	else:
		tmp = math.expm1(-(-x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.01)
		tmp = expm1(Float64(log(2.0) - log1p(exp(Float64(-2.0 * x)))));
	else
		tmp = expm1(Float64(-Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.01], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[(Exp[(-(-x))] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.0100000000000000002

   1. Initial program 99.7%

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

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-to-exp N/A

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

      7. lower-expm1.f64 N/A

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      13. metadata-eval 99.9

          \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot \color{blue}{-1}\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot -1\right)} \]
   5. Step-by-step derivation

      1. rem-log-exp N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. div-inv N/A

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

      8. inv-pow N/A

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

      9. clear-num N/A

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

      10. log-div N/A

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right) \]

      11. lower--.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right) \]

      12. lower-log.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\color{blue}{\log 2} - \log \left(1 + e^{-2 \cdot x}\right)\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\log 2 - \log \color{blue}{\left(1 + e^{-2 \cdot x}\right)}\right) \]

      14. lower-log1p.f64 99.9

          \[\leadsto \mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right) \]

   6. Applied rewrites 99.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right) \]

   if -0.0100000000000000002 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 34.3%

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

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-to-exp N/A

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

      7. lower-expm1.f64 N/A

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      13. metadata-eval 34.3

          \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot \color{blue}{-1}\right) \]

   4. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.7

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

   7. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{neg}\left(\left(-x\right)\right)}\right) \]

      4. lower-neg.f64 99.7

         \[\leadsto \mathsf{expm1}\left(\color{blue}{-\left(-x\right)}\right) \]

   9. Applied rewrites 99.7%

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

Alternative 3: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.01:\\ \;\;\;\;\mathsf{expm1}\left(-\log \left(\mathsf{fma}\left(e^{-2 \cdot x}, 0.5, 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(-\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.01)
   (expm1 (- (log (fma (exp (* -2.0 x)) 0.5 0.5))))
   (expm1 (- (- x)))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.01) {
		tmp = expm1(-log(fma(exp((-2.0 * x)), 0.5, 0.5)));
	} else {
		tmp = expm1(-(-x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.01)
		tmp = expm1(Float64(-log(fma(exp(Float64(-2.0 * x)), 0.5, 0.5))));
	else
		tmp = expm1(Float64(-Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.01], N[(Exp[(-N[Log[N[(N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision])] - 1), $MachinePrecision], N[(Exp[(-(-x))] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.0100000000000000002

   1. Initial program 99.7%

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

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-to-exp N/A

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

      7. lower-expm1.f64 N/A

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      13. metadata-eval 99.9

          \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot \color{blue}{-1}\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot -1\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{neg}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)\right)}\right) \]

      4. lower-neg.f64 99.9

         \[\leadsto \mathsf{expm1}\left(\color{blue}{-\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}\right) \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 99.9

          \[\leadsto \mathsf{expm1}\left(-\log \color{blue}{\left(\mathsf{fma}\left(e^{-2 \cdot x}, 0.5, 0.5\right)\right)}\right) \]

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(-\log \left(\mathsf{fma}\left(e^{-2 \cdot x}, 0.5, 0.5\right)\right)\right)} \]

   if -0.0100000000000000002 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 34.3%

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

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-to-exp N/A

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

      7. lower-expm1.f64 N/A

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      13. metadata-eval 34.3

          \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot \color{blue}{-1}\right) \]

   4. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.7

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

   7. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{neg}\left(\left(-x\right)\right)}\right) \]

      4. lower-neg.f64 99.7

         \[\leadsto \mathsf{expm1}\left(\color{blue}{-\left(-x\right)}\right) \]

   9. Applied rewrites 99.7%

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

Alternative 4: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, \mathsf{expm1}\left(-2 \cdot x\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(-\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.01)
   (fma (/ 2.0 (expm1 (* x -4.0))) (expm1 (* -2.0 x)) -1.0)
   (expm1 (- (- x)))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.01) {
		tmp = fma((2.0 / expm1((x * -4.0))), expm1((-2.0 * x)), -1.0);
	} else {
		tmp = expm1(-(-x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.01)
		tmp = fma(Float64(2.0 / expm1(Float64(x * -4.0))), expm1(Float64(-2.0 * x)), -1.0);
	else
		tmp = expm1(Float64(-Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.01], N[(N[(2.0 / N[(Exp[N[(x * -4.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision] + -1.0), $MachinePrecision], N[(Exp[(-(-x))] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.0100000000000000002

   1. Initial program 99.7%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-/r/ N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   if -0.0100000000000000002 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 34.3%

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

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-to-exp N/A

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

      7. lower-expm1.f64 N/A

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      13. metadata-eval 34.3

          \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot \color{blue}{-1}\right) \]

   4. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.7

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

   7. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{neg}\left(\left(-x\right)\right)}\right) \]

      4. lower-neg.f64 99.7

         \[\leadsto \mathsf{expm1}\left(\color{blue}{-\left(-x\right)}\right) \]

   9. Applied rewrites 99.7%

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

Alternative 5: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.3333333333333333, 2\right), -2\right), 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 2e-42)
   (+ (/ 2.0 (fma x (fma x (fma x -1.3333333333333333 2.0) -2.0) 2.0)) -1.0)
   (fma
    (fma (* x x) 0.13333333333333333 -0.3333333333333333)
    (* x (* x x))
    x)))

C

double code(double x, double y) {
	double tmp;
	if ((2.0 / (1.0 + exp((-2.0 * x)))) <= 2e-42) {
		tmp = (2.0 / fma(x, fma(x, fma(x, -1.3333333333333333, 2.0), -2.0), 2.0)) + -1.0;
	} else {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), (x * (x * x)), x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) <= 2e-42)
		tmp = Float64(Float64(2.0 / fma(x, fma(x, fma(x, -1.3333333333333333, 2.0), -2.0), 2.0)) + -1.0);
	else
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), Float64(x * Float64(x * x)), x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-42], N[(N[(2.0 / N[(x * N[(x * N[(x * -1.3333333333333333 + 2.0), $MachinePrecision] + -2.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) < 2.00000000000000008e-42

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 2.00000000000000008e-42 < (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))))

   1. Initial program 36.2%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.3333333333333333, 2\right), -2\right), 2\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.01:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(-\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.01)
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (expm1 (- (- x)))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.01) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = expm1(-(-x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.01) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = Math.expm1(-(-x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.01:
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	else:
		tmp = math.expm1(-(-x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.01)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	else
		tmp = expm1(Float64(-Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.01], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(Exp[(-(-x))] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.0100000000000000002

   1. Initial program 99.7%

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

   if -0.0100000000000000002 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 34.3%

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

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-to-exp N/A

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

      7. lower-expm1.f64 N/A

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

      13. metadata-eval 34.3

          \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot \color{blue}{-1}\right) \]

   4. Applied rewrites 34.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot -1\right)} \]

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.7

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

   7. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{neg}\left(\left(-x\right)\right)}\right) \]

      4. lower-neg.f64 99.7

         \[\leadsto \mathsf{expm1}\left(\color{blue}{-\left(-x\right)}\right) \]

   9. Applied rewrites 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.01:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(-\left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 36.2%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 75.6%

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

Alternative 8: 74.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(-\left(-x\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (expm1 (- (- x))))

C

double code(double x, double y) {
	return expm1(-(-x));
}

Java

public static double code(double x, double y) {
	return Math.expm1(-(-x));
}

Python

def code(x, y):
	return math.expm1(-(-x))

Julia

function code(x, y)
	return expm1(Float64(-Float64(-x)))
end

Wolfram

code[x_, y_] := N[(Exp[(-(-x))] - 1), $MachinePrecision]

Derivation

1. Initial program 50.4%

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

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

   3. clear-num N/A

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

   4. inv-pow N/A

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

   5. metadata-eval N/A

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

   6. pow-to-exp N/A

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

   7. lower-expm1.f64 N/A

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)}\right) \]

   9. lower-log.f64 N/A

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\frac{1 + e^{-2 \cdot x}}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

   10. div-inv N/A

       \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \frac{1}{2}\right)} \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right) \]

   13. metadata-eval 50.4

       \[\leadsto \mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot \color{blue}{-1}\right) \]

4. Applied rewrites 50.4%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\log \left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right) \cdot -1\right)} \]

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 76.2

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

7. Applied rewrites 76.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. mul-1-neg N/A

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{neg}\left(\left(-x\right)\right)}\right) \]

   4. lower-neg.f64 76.2

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-\left(-x\right)}\right) \]

9. Applied rewrites 76.2%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{-\left(-x\right)}\right) \]
10. Add Preprocessing

Alternative 9: 74.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.3333333333333333, 2\right), -2\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) 2e-13)
   (fma (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x (* x x)) x)
   (+ (/ 2.0 (* x (fma x (fma x -1.3333333333333333 2.0) -2.0))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= 2e-13) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = (2.0 / (x * fma(x, fma(x, -1.3333333333333333, 2.0), -2.0))) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= 2e-13)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = Float64(Float64(2.0 / Float64(x * fma(x, fma(x, -1.3333333333333333, 2.0), -2.0))) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-13], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(2.0 / N[(x * N[(x * N[(x * -1.3333333333333333 + 2.0), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 36.2%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.3333333333333333, 2\right), -2\right)} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot \mathsf{fma}\left(x, -1.3333333333333333, 2\right)\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) 2e-13)
   (fma (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x (* x x)) x)
   (+ (/ 2.0 (* x (* x (fma x -1.3333333333333333 2.0)))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= 2e-13) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = (2.0 / (x * (x * fma(x, -1.3333333333333333, 2.0)))) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= 2e-13)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = Float64(Float64(2.0 / Float64(x * Float64(x * fma(x, -1.3333333333333333, 2.0)))) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-13], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(2.0 / N[(x * N[(x * N[(x * -1.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 36.2%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot \mathsf{fma}\left(x, -1.3333333333333333, 2\right)\right)} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(-1.3333333333333333 \cdot \left(x \cdot x\right)\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) 2e-13)
   (fma (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x (* x x)) x)
   (+ (/ 2.0 (* x (* -1.3333333333333333 (* x x)))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= 2e-13) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = (2.0 / (x * (-1.3333333333333333 * (x * x)))) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= 2e-13)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = Float64(Float64(2.0 / Float64(x * Float64(-1.3333333333333333 * Float64(x * x)))) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-13], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(2.0 / N[(x * N[(-1.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 36.2%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(-1.3333333333333333 \cdot \left(x \cdot x\right)\right)} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, -2 + \left(x + x\right), 2\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) 2e-13)
   (fma (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x (* x x)) x)
   (+ (/ 2.0 (fma x (+ -2.0 (+ x x)) 2.0)) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= 2e-13) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = (2.0 / fma(x, (-2.0 + (x + x)), 2.0)) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= 2e-13)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = Float64(Float64(2.0 / fma(x, Float64(-2.0 + Float64(x + x)), 2.0)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-13], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(2.0 / N[(x * N[(-2.0 + N[(x + x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 36.2%

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

   3. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. count-2 N/A

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

      8. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, -2 + \left(x + x\right), 2\right)} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.7% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. count-2 N/A

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

      8. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if -1 < x

   1. Initial program 36.2%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

4. Final simplification 75.8%

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

Alternative 14: 74.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[x, -1.2], N[(N[(2.0 / N[(x * N[(x * 2.0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $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. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 98.5%

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

   if -1.19999999999999996 < x

   1. Initial program 36.2%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

4. Final simplification 75.8%

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

Alternative 15: 49.8% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 54.0

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

5. Applied rewrites 54.0%

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

Alternative 16: 6.6% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 6.5

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

5. Applied rewrites 6.5%

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

6. Final simplification 6.5%

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

Alternative 17: 4.2% accurate, 30.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.5%

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

6. Final simplification 4.5%

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

Reproduce

herbie shell --seed 2024221 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))