Logistic function from Lakshay Garg

Percentage Accurate: 54.9% → 99.3%

Time: 8.2s

Alternatives: 10

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + {\left(e^{x}\right)}^{-2}\\ t_1 := 1 + e^{-2 \cdot x}\\ t_2 := \frac{2}{t\_1}\\ t_3 := {t\_0}^{-2}\\ \mathbf{if}\;-2 \cdot x \leq -10000000:\\ \;\;\;\;t\_2 - 1\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{8}{t\_0}, \frac{t\_3}{\mathsf{fma}\left(4, t\_3, \frac{2}{t\_0} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {t\_1}^{-2}, t\_2 - -1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (pow (exp x) -2.0)))
        (t_1 (+ 1.0 (exp (* -2.0 x))))
        (t_2 (/ 2.0 t_1))
        (t_3 (pow t_0 -2.0)))
   (if (<= (* -2.0 x) -10000000.0)
     (- t_2 1.0)
     (if (<= (* -2.0 x) 1e-7)
       (fma
        (* (fma 0.13333333333333333 (* x x) -0.3333333333333333) (* x x))
        x
        x)
       (fma
        (/ 8.0 t_0)
        (/ t_3 (fma 4.0 t_3 (- (/ 2.0 t_0) -1.0)))
        (/ -1.0 (fma 4.0 (pow t_1 -2.0) (- t_2 -1.0))))))))

C

double code(double x) {
	double t_0 = 1.0 + pow(exp(x), -2.0);
	double t_1 = 1.0 + exp((-2.0 * x));
	double t_2 = 2.0 / t_1;
	double t_3 = pow(t_0, -2.0);
	double tmp;
	if ((-2.0 * x) <= -10000000.0) {
		tmp = t_2 - 1.0;
	} else if ((-2.0 * x) <= 1e-7) {
		tmp = fma((fma(0.13333333333333333, (x * x), -0.3333333333333333) * (x * x)), x, x);
	} else {
		tmp = fma((8.0 / t_0), (t_3 / fma(4.0, t_3, ((2.0 / t_0) - -1.0))), (-1.0 / fma(4.0, pow(t_1, -2.0), (t_2 - -1.0))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + (exp(x) ^ -2.0))
	t_1 = Float64(1.0 + exp(Float64(-2.0 * x)))
	t_2 = Float64(2.0 / t_1)
	t_3 = t_0 ^ -2.0
	tmp = 0.0
	if (Float64(-2.0 * x) <= -10000000.0)
		tmp = Float64(t_2 - 1.0);
	elseif (Float64(-2.0 * x) <= 1e-7)
		tmp = fma(Float64(fma(0.13333333333333333, Float64(x * x), -0.3333333333333333) * Float64(x * x)), x, x);
	else
		tmp = fma(Float64(8.0 / t_0), Float64(t_3 / fma(4.0, t_3, Float64(Float64(2.0 / t_0) - -1.0))), Float64(-1.0 / fma(4.0, (t_1 ^ -2.0), Float64(t_2 - -1.0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$0, -2.0], $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -10000000.0], N[(t$95$2 - 1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-7], N[(N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(8.0 / t$95$0), $MachinePrecision] * N[(t$95$3 / N[(4.0 * t$95$3 + N[(N[(2.0 / t$95$0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(4.0 * N[Power[t$95$1, -2.0], $MachinePrecision] + N[(t$95$2 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e7

   1. Initial program 100.0%

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

   if -1e7 < (*.f64 #s(literal -2 binary64) x) < 9.9999999999999995e-8

   1. Initial program 6.8%

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

   4. Applied rewrites 6.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if 9.9999999999999995e-8 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{8}{1 + {\left(e^{x}\right)}^{-2}}, \frac{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}\right)} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{8}{1 + {\left(e^{x}\right)}^{-2}}, \frac{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + \color{blue}{{\left(e^{x}\right)}^{-2}}} - -1\right)}\right) \]

      2. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{8}{1 + {\left(e^{x}\right)}^{-2}}, \frac{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\color{blue}{\left(e^{x}\right)}}^{-2}} - -1\right)}\right) \]

      3. pow-exp N/A

         \[\leadsto \mathsf{fma}\left(\frac{8}{1 + {\left(e^{x}\right)}^{-2}}, \frac{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + \color{blue}{e^{x \cdot -2}}} - -1\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{8}{1 + {\left(e^{x}\right)}^{-2}}, \frac{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + e^{\color{blue}{-2 \cdot x}}} - -1\right)}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{8}{1 + {\left(e^{x}\right)}^{-2}}, \frac{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + \color{blue}{e^{-2 \cdot x}}} - -1\right)}\right) \]

      6. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(\frac{8}{1 + {\left(e^{x}\right)}^{-2}}, \frac{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + e^{\color{blue}{-2 \cdot x}}} - -1\right)}\right) \]

   8. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{8}{1 + {\left(e^{x}\right)}^{-2}}, \frac{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + \color{blue}{e^{-2 \cdot x}}} - -1\right)}\right) \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{8}{1 + {\left(e^{x}\right)}^{-2}}, \frac{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {\left(1 + \color{blue}{{\left(e^{x}\right)}^{-2}}\right)}^{-2}, \frac{2}{1 + e^{-2 \cdot x}} - -1\right)}\right) \]

      2. lift-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{8}{1 + {\left(e^{x}\right)}^{-2}}, \frac{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {\left(1 + {\color{blue}{\left(e^{x}\right)}}^{-2}\right)}^{-2}, \frac{2}{1 + e^{-2 \cdot x}} - -1\right)}\right) \]

      3. pow-exp N/A

         \[\leadsto \mathsf{fma}\left(\frac{8}{1 + {\left(e^{x}\right)}^{-2}}, \frac{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {\left(1 + \color{blue}{e^{x \cdot -2}}\right)}^{-2}, \frac{2}{1 + e^{-2 \cdot x}} - -1\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{8}{1 + {\left(e^{x}\right)}^{-2}}, \frac{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {\left(1 + e^{\color{blue}{-2 \cdot x}}\right)}^{-2}, \frac{2}{1 + e^{-2 \cdot x}} - -1\right)}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{8}{1 + {\left(e^{x}\right)}^{-2}}, \frac{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {\left(1 + e^{\color{blue}{-2 \cdot x}}\right)}^{-2}, \frac{2}{1 + e^{-2 \cdot x}} - -1\right)}\right) \]

      6. lift-exp.f64 100.0

         \[\leadsto \mathsf{fma}\left(\frac{8}{1 + {\left(e^{x}\right)}^{-2}}, \frac{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {\left(1 + \color{blue}{e^{-2 \cdot x}}\right)}^{-2}, \frac{2}{1 + e^{-2 \cdot x}} - -1\right)}\right) \]

   10. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\frac{8}{1 + {\left(e^{x}\right)}^{-2}}, \frac{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}, \frac{2}{1 + {\left(e^{x}\right)}^{-2}} - -1\right)}, \frac{-1}{\mathsf{fma}\left(4, {\left(1 + \color{blue}{e^{-2 \cdot x}}\right)}^{-2}, \frac{2}{1 + e^{-2 \cdot x}} - -1\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10000000 \lor \neg \left(-2 \cdot x \leq 10^{-7}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= (* -2.0 x) -10000000.0) (not (<= (* -2.0 x) 1e-7)))
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (fma
    (* (fma 0.13333333333333333 (* x x) -0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (((-2.0 * x) <= -10000000.0) || !((-2.0 * x) <= 1e-7)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else {
		tmp = fma((fma(0.13333333333333333, (x * x), -0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -10000000.0) || !(Float64(-2.0 * x) <= 1e-7))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	else
		tmp = fma(Float64(fma(0.13333333333333333, Float64(x * x), -0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -10000000.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-7]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e7 or 9.9999999999999995e-8 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -1e7 < (*.f64 #s(literal -2 binary64) x) < 9.9999999999999995e-8

   1. Initial program 6.8%

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

   4. Applied rewrites 6.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10000000 \lor \neg \left(-2 \cdot x \leq 10^{-7}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(-1.3333333333333333 * x + 2.0), $MachinePrecision] * x), $MachinePrecision] * x), $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 40.0%

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

   4. Applied rewrites 40.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.8

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 65.8%

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

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.5%

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

Alternative 4: 74.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(2.0 / N[(N[(N[(-1.3333333333333333 * x), $MachinePrecision] * x), $MachinePrecision] * x), $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 40.0%

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

   4. Applied rewrites 40.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.8

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 65.8%

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

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 99.5%

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

Alternative 5: 74.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(N[(0.13333333333333333 * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(2.0 / N[(N[(2.0 * x + -2.0), $MachinePrecision] * x + 2.0), $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 40.0%

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

   4. Applied rewrites 40.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 65.8

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 65.8%

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

   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

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(2 \cdot x - 2\right)}} - 1 \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

Alternative 6: 29.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 0.2:\\ \;\;\;\;\left(1 + x\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{-2 \cdot x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* -2.0 x) 0.2) (- (+ 1.0 x) 1.0) (- (/ 2.0 (* -2.0 x)) 1.0)))

C

double code(double x) {
	double tmp;
	if ((-2.0 * x) <= 0.2) {
		tmp = (1.0 + x) - 1.0;
	} else {
		tmp = (2.0 / (-2.0 * x)) - 1.0;
	}
	return tmp;
}

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if (-2.0 * x) <= 0.2:
		tmp = (1.0 + x) - 1.0
	else:
		tmp = (2.0 / (-2.0 * x)) - 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(-2.0 * x) <= 0.2)
		tmp = Float64(Float64(1.0 + x) - 1.0);
	else
		tmp = Float64(Float64(2.0 / Float64(-2.0 * x)) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((-2.0 * x) <= 0.2)
		tmp = (1.0 + x) - 1.0;
	else
		tmp = (2.0 / (-2.0 * x)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 40.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 6.4

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

   5. Applied rewrites 6.4%

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

   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. +-commutative N/A

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

      2. lower-fma.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.0%

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

Alternative 7: 28.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, -2\right), x, 2\right)} - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 2.0 (fma (fma 2.0 x -2.0) x 2.0)) 1.0))

C

double code(double x) {
	return (2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0;
}

Julia

function code(x)
	return Float64(Float64(2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0)
end

Wolfram

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(2 \cdot x - 2\right)}} - 1 \]
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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 25.9

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

5. Applied rewrites 25.9%

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

Alternative 8: 28.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(-2, x, 2\right)} - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 2.0 (fma -2.0 x 2.0)) 1.0))

C

double code(double x) {
	return (2.0 / fma(-2.0, x, 2.0)) - 1.0;
}

Julia

function code(x)
	return Float64(Float64(2.0 / fma(-2.0, x, 2.0)) - 1.0)
end

Wolfram

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

Derivation

1. Initial program 53.4%

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

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

   2. lower-fma.f64 25.7

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

5. Applied rewrites 25.7%

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

Alternative 9: 6.6% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in x around 0

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

   1. lower-+.f64 6.1

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

5. Applied rewrites 6.1%

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

Alternative 10: 4.2% accurate, 30.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 - 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.3%

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

Reproduce

herbie shell --seed 2024327 
(FPCore (x)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))