Logistic function from Lakshay Garg

Percentage Accurate: 53.2% → 99.9%

Time: 10.9s

Alternatives: 11

Speedup: 5.3×

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: 53.2% 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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ t_1 := 1 + t\_0\\ t_2 := \frac{2}{t\_1}\\ t_3 := 1 + t\_2\\ t_4 := t\_1 \cdot 0.5\\ \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;t\_2 + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{t\_4}^{-3}}{\mathsf{fma}\left(64, {t\_1}^{-6}, {t\_3}^{3}\right)}, \mathsf{fma}\left(t\_3, t\_3 - {t\_4}^{-2}, {t\_4}^{-4}\right), \frac{1}{\frac{2}{-1 - t\_0} - \mathsf{fma}\left(4, {t\_1}^{-2}, 1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x)))
        (t_1 (+ 1.0 t_0))
        (t_2 (/ 2.0 t_1))
        (t_3 (+ 1.0 t_2))
        (t_4 (* t_1 0.5)))
   (if (<= (* -2.0 x) -20.0)
     (+ t_2 -1.0)
     (if (<= (* -2.0 x) 0.002)
       (fma
        (fma (* x x) 0.13333333333333333 -0.3333333333333333)
        (* x (* x x))
        x)
       (fma
        (/ (pow t_4 -3.0) (fma 64.0 (pow t_1 -6.0) (pow t_3 3.0)))
        (fma t_3 (- t_3 (pow t_4 -2.0)) (pow t_4 -4.0))
        (/ 1.0 (- (/ 2.0 (- -1.0 t_0)) (fma 4.0 (pow t_1 -2.0) 1.0))))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double t_1 = 1.0 + t_0;
	double t_2 = 2.0 / t_1;
	double t_3 = 1.0 + t_2;
	double t_4 = t_1 * 0.5;
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = t_2 + -1.0;
	} else if ((-2.0 * x) <= 0.002) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = fma((pow(t_4, -3.0) / fma(64.0, pow(t_1, -6.0), pow(t_3, 3.0))), fma(t_3, (t_3 - pow(t_4, -2.0)), pow(t_4, -4.0)), (1.0 / ((2.0 / (-1.0 - t_0)) - fma(4.0, pow(t_1, -2.0), 1.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(2.0 / t_1)
	t_3 = Float64(1.0 + t_2)
	t_4 = Float64(t_1 * 0.5)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20.0)
		tmp = Float64(t_2 + -1.0);
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = fma(Float64((t_4 ^ -3.0) / fma(64.0, (t_1 ^ -6.0), (t_3 ^ 3.0))), fma(t_3, Float64(t_3 - (t_4 ^ -2.0)), (t_4 ^ -4.0)), Float64(1.0 / Float64(Float64(2.0 / Float64(-1.0 - t_0)) - fma(4.0, (t_1 ^ -2.0), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 * 0.5), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -20.0], N[(t$95$2 + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[Power[t$95$4, -3.0], $MachinePrecision] / N[(64.0 * N[Power[t$95$1, -6.0], $MachinePrecision] + N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[(t$95$3 - N[Power[t$95$4, -2.0], $MachinePrecision]), $MachinePrecision] + N[Power[t$95$4, -4.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[(2.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] - N[(4.0 * N[Power[t$95$1, -2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -20

   1. Initial program 100.0%

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

   if -20 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 9.9%

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

   3. Taylor expanded in x around 0

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

      1. 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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

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

   1. Initial program 99.9%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-3}}{\mathsf{fma}\left(64, {\left(1 + e^{-2 \cdot x}\right)}^{-6}, {\left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}, \mathsf{fma}\left(1 + \frac{2}{1 + e^{-2 \cdot x}}, \left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) - {\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-2}, {\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-4}\right), -\frac{1}{\frac{2}{1 + e^{-2 \cdot x}} + \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-3}}{\mathsf{fma}\left(64, {\left(1 + e^{-2 \cdot x}\right)}^{-6}, {\left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right)}^{3}\right)}, \mathsf{fma}\left(1 + \frac{2}{1 + e^{-2 \cdot x}}, \left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) - {\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-2}, {\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-4}\right), \frac{1}{\frac{2}{-1 - e^{-2 \cdot x}} - \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ t_1 := \frac{2}{t\_0}\\ t_2 := 1 + t\_1\\ \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;t\_1 + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{t\_0}^{-2}}{t\_2}, 4, \frac{-1}{t\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x)))) (t_1 (/ 2.0 t_0)) (t_2 (+ 1.0 t_1)))
   (if (<= (* -2.0 x) -20.0)
     (+ t_1 -1.0)
     (if (<= (* -2.0 x) 0.002)
       (fma
        (fma (* x x) 0.13333333333333333 -0.3333333333333333)
        (* x (* x x))
        x)
       (fma (/ (pow t_0 -2.0) t_2) 4.0 (/ -1.0 t_2))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + exp((-2.0 * x));
	double t_1 = 2.0 / t_0;
	double t_2 = 1.0 + t_1;
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = t_1 + -1.0;
	} else if ((-2.0 * x) <= 0.002) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = fma((pow(t_0, -2.0) / t_2), 4.0, (-1.0 / t_2));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + exp(Float64(-2.0 * x)))
	t_1 = Float64(2.0 / t_0)
	t_2 = Float64(1.0 + t_1)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20.0)
		tmp = Float64(t_1 + -1.0);
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = fma(Float64((t_0 ^ -2.0) / t_2), 4.0, Float64(-1.0 / t_2));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -20.0], N[(t$95$1 + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[Power[t$95$0, -2.0], $MachinePrecision] / t$95$2), $MachinePrecision] * 4.0 + N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -20

   1. Initial program 100.0%

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

   if -20 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 9.9%

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

   3. Taylor expanded in x around 0

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

      1. 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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

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

   1. Initial program 99.9%

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

      1. flip-- N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. 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 egg-rr 100.0%

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

      1. *-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(1 + e^{-2 \cdot x}\right)}^{-2}}{1 + \frac{2}{1 + e^{-2 \cdot x}}}, 4, \frac{-1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;\frac{2}{1 + t\_0} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))))
   (if (<= (* -2.0 x) -20.0)
     (+ (/ 2.0 (+ 1.0 t_0)) -1.0)
     (if (<= (* -2.0 x) 0.002)
       (fma
        (fma (* x x) 0.13333333333333333 -0.3333333333333333)
        (* x (* x x))
        x)
       (expm1 (- (log 2.0) (log1p t_0)))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = (2.0 / (1.0 + t_0)) + -1.0;
	} else if ((-2.0 * x) <= 0.002) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = expm1((log(2.0) - log1p(t_0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + t_0)) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = expm1(Float64(log(2.0) - log1p(t_0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -20.0], N[(N[(2.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -20

   1. Initial program 100.0%

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

   if -20 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 9.9%

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

   3. Taylor expanded in x around 0

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

      1. 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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

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

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. metadata-eval N/A

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

      4. pow-to-exp N/A

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

      5. accelerator-lowering-expm1.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. div-inv N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-lowering-+.f64 N/A

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

      11. exp-lowering-exp.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. 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 egg-rr 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. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

      1. distribute-lft1-in N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

      6. neg-log N/A

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

      7. clear-num N/A

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

      8. log-div N/A

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

      9. --lowering--.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. accelerator-lowering-log1p.f64 N/A

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

      12. *-commutative N/A

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

      13. exp-lowering-exp.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 100.0

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

   9. Applied egg-rr 100.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;\frac{2}{1 + t\_0} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(-\log \left(\mathsf{fma}\left(t\_0, 0.5, 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))))
   (if (<= (* -2.0 x) -20.0)
     (+ (/ 2.0 (+ 1.0 t_0)) -1.0)
     (if (<= (* -2.0 x) 0.002)
       (fma
        (fma (* x x) 0.13333333333333333 -0.3333333333333333)
        (* x (* x x))
        x)
       (expm1 (- (log (fma t_0 0.5 0.5))))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = (2.0 / (1.0 + t_0)) + -1.0;
	} else if ((-2.0 * x) <= 0.002) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = expm1(-log(fma(t_0, 0.5, 0.5)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + t_0)) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = expm1(Float64(-log(fma(t_0, 0.5, 0.5))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -20.0], N[(N[(2.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(Exp[(-N[Log[N[(t$95$0 * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision])] - 1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -20

   1. Initial program 100.0%

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

   if -20 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 9.9%

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

   3. Taylor expanded in x around 0

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

      1. 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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

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

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. metadata-eval N/A

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

      4. pow-to-exp N/A

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

      5. accelerator-lowering-expm1.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. div-inv N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-lowering-+.f64 N/A

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

      11. exp-lowering-exp.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. 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 egg-rr 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. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-\log \left(\mathsf{fma}\left(e^{x \cdot -2}, 0.5, 0.5\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(-\log \left(\mathsf{fma}\left(e^{-2 \cdot x}, 0.5, 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)))
   (if (<= (* -2.0 x) -20.0)
     t_0
     (if (<= (* -2.0 x) 0.002)
       (fma
        (fma (* x x) 0.13333333333333333 -0.3333333333333333)
        (* x (* x x))
        x)
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 0.002) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20.0)
		tmp = t_0;
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -20.0], t$95$0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -20 or 2e-3 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -20 < (*.f64 #s(literal -2 binary64) x) < 2e-3

   1. Initial program 9.9%

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

   3. Taylor expanded in x around 0

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

      1. 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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;1\\ \mathbf{elif}\;-2 \cdot x \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -20.0)
   1.0
   (if (<= (* -2.0 x) 0.2)
     (fma
      (fma
       (* x x)
       (fma (* x x) -0.05396825396825397 0.13333333333333333)
       -0.3333333333333333)
      (* x (* x x))
      x)
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = 1.0;
	} else if ((-2.0 * x) <= 0.2) {
		tmp = fma(fma((x * x), fma((x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20.0)
		tmp = 1.0;
	elseif (Float64(-2.0 * x) <= 0.2)
		tmp = fma(fma(Float64(x * x), fma(Float64(x * x), -0.05396825396825397, 0.13333333333333333), -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -20.0], 1.0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.2], N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.05396825396825397 + 0.13333333333333333), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -20

   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. --lowering--.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. +-lowering-+.f64 1.6

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

   5. Simplified 1.6%

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

   7. Applied egg-rr 97.4%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 99.4%

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

   if -20 < (*.f64 #s(literal -2 binary64) x) < 0.20000000000000001

   1. Initial program 10.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.8%

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

   if 0.20000000000000001 < (*.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. --lowering--.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. +-lowering-+.f64 96.0

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

   5. Simplified 96.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

Alternative 7: 99.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;1\\ \mathbf{elif}\;-2 \cdot x \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -20.0)
   1.0
   (if (<= (* -2.0 x) 0.2)
     (fma
      (fma (* x x) 0.13333333333333333 -0.3333333333333333)
      (* x (* x x))
      x)
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = 1.0;
	} else if ((-2.0 * x) <= 0.2) {
		tmp = fma(fma((x * x), 0.13333333333333333, -0.3333333333333333), (x * (x * x)), x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20.0)
		tmp = 1.0;
	elseif (Float64(-2.0 * x) <= 0.2)
		tmp = fma(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333), Float64(x * Float64(x * x)), x);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -20.0], 1.0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.2], N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -20

   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. --lowering--.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. +-lowering-+.f64 1.6

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

   5. Simplified 1.6%

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

   7. Applied egg-rr 97.4%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 99.4%

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

   if -20 < (*.f64 #s(literal -2 binary64) x) < 0.20000000000000001

   1. Initial program 10.6%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 99.7

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

   5. Simplified 99.7%

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

   if 0.20000000000000001 < (*.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. --lowering--.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. +-lowering-+.f64 96.0

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

   5. Simplified 96.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

Alternative 8: 99.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;1\\ \mathbf{elif}\;-2 \cdot x \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -20.0)
   1.0
   (if (<= (* -2.0 x) 0.2) (fma -0.3333333333333333 (* x (* x x)) x) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = 1.0;
	} else if ((-2.0 * x) <= 0.2) {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20.0)
		tmp = 1.0;
	elseif (Float64(-2.0 * x) <= 0.2)
		tmp = fma(-0.3333333333333333, Float64(x * Float64(x * x)), x);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -20.0], 1.0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.2], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -20

   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. --lowering--.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. +-lowering-+.f64 1.6

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

   5. Simplified 1.6%

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

   7. Applied egg-rr 97.4%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 99.4%

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

   if -20 < (*.f64 #s(literal -2 binary64) x) < 0.20000000000000001

   1. Initial program 10.6%

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

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

      8. *-lowering-*.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. *-lowering-*.f64 99.3

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

   5. Simplified 99.3%

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

   if 0.20000000000000001 < (*.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. --lowering--.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. +-lowering-+.f64 96.0

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

   5. Simplified 96.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

Alternative 9: 99.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;1\\ \mathbf{elif}\;-2 \cdot x \leq 0.2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -20.0) 1.0 (if (<= (* -2.0 x) 0.2) x -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = 1.0;
	} else if ((-2.0 * x) <= 0.2) {
		tmp = x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((-2.0d0) * x) <= (-20.0d0)) then
        tmp = 1.0d0
    else if (((-2.0d0) * x) <= 0.2d0) then
        tmp = x
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = 1.0;
	} else if ((-2.0 * x) <= 0.2) {
		tmp = x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -20.0:
		tmp = 1.0
	elif (-2.0 * x) <= 0.2:
		tmp = x
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20.0)
		tmp = 1.0;
	elseif (Float64(-2.0 * x) <= 0.2)
		tmp = x;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -20.0)
		tmp = 1.0;
	elseif ((-2.0 * x) <= 0.2)
		tmp = x;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -20.0], 1.0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.2], x, -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -20

   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. --lowering--.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. +-lowering-+.f64 1.6

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

   5. Simplified 1.6%

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

   7. Applied egg-rr 97.4%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 99.4%

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

   if -20 < (*.f64 #s(literal -2 binary64) x) < 0.20000000000000001

   1. Initial program 10.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.1%

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

   if 0.20000000000000001 < (*.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. --lowering--.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. +-lowering-+.f64 96.0

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

   5. Simplified 96.0%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 100.0%

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

Alternative 10: 51.6% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5 \cdot 10^{-311}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* -2.0 x) -5e-311) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5e-311) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((-2.0d0) * x) <= (-5d-311)) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5e-311) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -5e-311:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5e-311)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -5e-311)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5e-311], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -5.00000000000023e-311

   1. Initial program 56.4%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. --lowering--.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. +-lowering-+.f64 5.4

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

   5. Simplified 5.4%

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

   7. Applied egg-rr 52.6%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 53.0%

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

   if -5.00000000000023e-311 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 49.6%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. --lowering--.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. +-lowering-+.f64 46.9

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

   5. Simplified 46.9%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 47.3%

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

Alternative 11: 27.1% accurate, 123.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 52.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\color{blue}{2 + -2 \cdot x}} - 1 \]
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. --lowering--.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. +-lowering-+.f64 27.3

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

5. Simplified 27.3%

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

6. Taylor expanded in x around inf

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

8. Simplified 25.8%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))