Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 15.0s

Alternatives: 22

Speedup: 1.1×

Specification

\[0 \leq s \land s \leq 1.0651631\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ \frac{{\left(\frac{1}{e^{t\_0}} + 1\right)}^{-2} \cdot e^{-t\_0}}{s} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) s)))
   (/ (* (pow (+ (/ 1.0 (exp t_0)) 1.0) -2.0) (exp (- t_0))) s)))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / s;
	return (powf(((1.0f / expf(t_0)) + 1.0f), -2.0f) * expf(-t_0)) / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = abs(x) / s
    code = ((((1.0e0 / exp(t_0)) + 1.0e0) ** (-2.0e0)) * exp(-t_0)) / s
end function

Julia

function code(x, s)
	t_0 = Float32(abs(x) / s)
	return Float32(Float32((Float32(Float32(Float32(1.0) / exp(t_0)) + Float32(1.0)) ^ Float32(-2.0)) * exp(Float32(-t_0))) / s)
end

MATLAB

function tmp = code(x, s)
	t_0 = abs(x) / s;
	tmp = ((((single(1.0) / exp(t_0)) + single(1.0)) ^ single(-2.0)) * exp(-t_0)) / s;
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

4. Applied egg-rr 99.2%

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

   1. lift-fabs.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lower-/.f32 N/A

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

   5. lower-exp.f32 99.2

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

6. Applied egg-rr 99.2%

   \[\leadsto \frac{{\left(\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}} + 1\right)}^{-2} \cdot e^{-\frac{\left|x\right|}{s}}}{s} \]
7. Add Preprocessing

Alternative 2: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0.05000000074505806:\\ \;\;\;\;\frac{t\_0}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 0.05000000074505806)
     (/ t_0 s)
     (/ (fma (/ x s) (/ (* x -0.0625) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 0.05000000074505806f) {
		tmp = t_0 / s;
	} else {
		tmp = fmaf((x / s), ((x * -0.0625f) / s), 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(0.05000000074505806))
		tmp = Float32(t_0 / s);
	else
		tmp = Float32(fma(Float32(x / s), Float32(Float32(x * Float32(-0.0625)) / s), Float32(0.25)) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.0500000007

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 99.2%

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

      1. lift-fabs.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift-pow.f32 N/A

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

      7. lift-fabs.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-neg.f32 N/A

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

      10. lift-exp.f32 N/A

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

      11. lift-*.f32 N/A

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

   6. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right), \frac{\left|x\right|}{-s}\right)}}{s}} \]

   7. Taylor expanded in s around 0

      \[\leadsto \frac{e^{\color{blue}{-1 \cdot \frac{\left|x\right|}{s}}}}{s} \]
   8. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}{s} \]

      2. lower-neg.f32 N/A

         \[\leadsto \frac{e^{\color{blue}{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}}{s} \]

      3. lower-/.f32 N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(\color{blue}{\frac{\left|x\right|}{s}}\right)}}{s} \]

      4. lower-fabs.f32 99.0

         \[\leadsto \frac{e^{-\frac{\color{blue}{\left|x\right|}}{s}}}{s} \]

   9. Simplified 99.0%

      \[\leadsto \frac{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}{s} \]

   if 0.0500000007 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 99.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]

   5. Simplified 68.2%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot s}}{s}} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{16}}{s \cdot s}}{s} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{16}}}{s \cdot s}}{s} \]

      3. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{\color{blue}{s \cdot s}}}{s} \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s}}}{s} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s} + \frac{1}{4}}}{s} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s}} + \frac{1}{4}}{s} \]

      7. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{16}}}{s \cdot s} + \frac{1}{4}}{s} \]

      8. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{16}}{s \cdot s} + \frac{1}{4}}{s} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \frac{-1}{16}\right)}}{s \cdot s} + \frac{1}{4}}{s} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{\frac{x \cdot \left(x \cdot \frac{-1}{16}\right)}{\color{blue}{s \cdot s}} + \frac{1}{4}}{s} \]

      11. times-frac N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{s} \cdot \frac{x \cdot \frac{-1}{16}}{s}} + \frac{1}{4}}{s} \]

      12. lower-fma.f32 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot \frac{-1}{16}}{s}, \frac{1}{4}\right)}}{s} \]

      13. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x}{s}}, \frac{x \cdot \frac{-1}{16}}{s}, \frac{1}{4}\right)}{s} \]

      14. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{x \cdot \frac{-1}{16}}{s}}, \frac{1}{4}\right)}{s} \]

      15. lower-*.f32 89.0

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \frac{\color{blue}{x \cdot -0.0625}}{s}, 0.25\right)}{s} \]

   7. Applied egg-rr 89.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(e^{-\frac{\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \leq 0.05000000074505806:\\ \;\;\;\;\frac{e^{-\frac{\left|x\right|}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 30.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0:\\ \;\;\;\;\frac{1}{\left|x\right| \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 0.0)
     (/ 1.0 (* (fabs x) -2.0))
     (/ 0.25 s))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 0.0f) {
		tmp = 1.0f / (fabsf(x) * -2.0f);
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp(-(abs(x) / s))
    t_1 = t_0 + 1.0e0
    if ((t_0 / (t_1 * (s * t_1))) <= 0.0e0) then
        tmp = 1.0e0 / (abs(x) * (-2.0e0))
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(0.0))
		tmp = Float32(Float32(1.0) / Float32(abs(x) * Float32(-2.0)));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp(-(abs(x) / s));
	t_1 = t_0 + single(1.0);
	tmp = single(0.0);
	if ((t_0 / (t_1 * (s * t_1))) <= single(0.0))
		tmp = single(1.0) / (abs(x) * single(-2.0));
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 0.0

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

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

   5. Simplified 99.2%

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

   6. Taylor expanded in s around inf

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

      1. lower-*.f32 N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fabs.f32 99.2

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

   8. Simplified 99.2%

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

   9. Taylor expanded in s around inf

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

   11. Simplified 33.2%

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

   12. Taylor expanded in s around 0

       \[\leadsto \frac{1}{\color{blue}{-2 \cdot \left|x\right|}} \]
   13. Step-by-step derivation

       1. lower-*.f32 N/A

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

       2. lower-fabs.f32 8.9

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

   14. Simplified 8.9%

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

   if 0.0 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 98.9%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 85.1

         \[\leadsto \color{blue}{\frac{0.25}{s}} \]

   5. Simplified 85.1%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(e^{-\frac{\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \leq 0:\\ \;\;\;\;\frac{1}{\left|x\right| \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\mathsf{fma}\left(\frac{-1}{s}, \left|x\right|, -2 \cdot \mathsf{log1p}\left(e^{-\frac{\left|x\right|}{s}}\right)\right)}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  (exp (fma (/ -1.0 s) (fabs x) (* -2.0 (log1p (exp (- (/ (fabs x) s)))))))
  s))

C

float code(float x, float s) {
	return expf(fmaf((-1.0f / s), fabsf(x), (-2.0f * log1pf(expf(-(fabsf(x) / s)))))) / s;
}

Julia

function code(x, s)
	return Float32(exp(fma(Float32(Float32(-1.0) / s), abs(x), Float32(Float32(-2.0) * log1p(exp(Float32(-Float32(abs(x) / s))))))) / s)
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

4. Applied egg-rr 99.2%

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

   1. lift-fabs.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-+.f32 N/A

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

   6. lift-pow.f32 N/A

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

   7. lift-fabs.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift-*.f32 N/A

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

6. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right), \frac{\left|x\right|}{-s}\right)}}{s}} \]
7. Step-by-step derivation

   1. lift-fabs.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-log1p.f32 N/A

      \[\leadsto \frac{e^{-2 \cdot \color{blue}{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{\mathsf{neg}\left(s\right)}}\right)} + \frac{\left|x\right|}{\mathsf{neg}\left(s\right)}}}{s} \]

   6. lift-fabs.f32 N/A

      \[\leadsto \frac{e^{-2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{\mathsf{neg}\left(s\right)}}\right) + \frac{\color{blue}{\left|x\right|}}{\mathsf{neg}\left(s\right)}}}{s} \]

   7. lift-neg.f32 N/A

      \[\leadsto \frac{e^{-2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{\mathsf{neg}\left(s\right)}}\right) + \frac{\left|x\right|}{\color{blue}{\mathsf{neg}\left(s\right)}}}}{s} \]

   8. lift-/.f32 N/A

      \[\leadsto \frac{e^{-2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{\mathsf{neg}\left(s\right)}}\right) + \color{blue}{\frac{\left|x\right|}{\mathsf{neg}\left(s\right)}}}}{s} \]

   9. +-commutative N/A

      \[\leadsto \frac{e^{\color{blue}{\frac{\left|x\right|}{\mathsf{neg}\left(s\right)} + -2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{\mathsf{neg}\left(s\right)}}\right)}}}{s} \]

   10. lift-/.f32 N/A

       \[\leadsto \frac{e^{\color{blue}{\frac{\left|x\right|}{\mathsf{neg}\left(s\right)}} + -2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{\mathsf{neg}\left(s\right)}}\right)}}{s} \]

   11. clear-num N/A

       \[\leadsto \frac{e^{\color{blue}{\frac{1}{\frac{\mathsf{neg}\left(s\right)}{\left|x\right|}}} + -2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{\mathsf{neg}\left(s\right)}}\right)}}{s} \]

   12. associate-/r/ N/A

       \[\leadsto \frac{e^{\color{blue}{\frac{1}{\mathsf{neg}\left(s\right)} \cdot \left|x\right|} + -2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{\mathsf{neg}\left(s\right)}}\right)}}{s} \]

   13. lower-fma.f32 N/A

       \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(s\right)}, \left|x\right|, -2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{\mathsf{neg}\left(s\right)}}\right)\right)}}}{s} \]

8. Applied egg-rr 99.2%

   \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(\frac{-1}{s}, \left|x\right|, -2 \cdot \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)\right)}}}{s} \]

9. Final simplification 99.2%

   \[\leadsto \frac{e^{\mathsf{fma}\left(\frac{-1}{s}, \left|x\right|, -2 \cdot \mathsf{log1p}\left(e^{-\frac{\left|x\right|}{s}}\right)\right)}}{s} \]
10. Add Preprocessing

Alternative 5: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ \frac{t\_0}{s \cdot {\left(t\_0 + 1\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s))))) (/ t_0 (* s (pow (+ t_0 1.0) 2.0)))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	return t_0 / (s * powf((t_0 + 1.0f), 2.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp(-(abs(x) / s))
    code = t_0 / (s * ((t_0 + 1.0e0) ** 2.0e0))
end function

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	return Float32(t_0 / Float32(s * (Float32(t_0 + Float32(1.0)) ^ Float32(2.0))))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp(-(abs(x) / s));
	tmp = t_0 / (s * ((t_0 + single(1.0)) ^ single(2.0)));
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-fabs.f32 N/A

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

   2. remove-double-neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. frac-2neg N/A

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

   6. frac-2neg N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. lift-fabs.f32 N/A

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

   11. remove-double-neg N/A

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

   12. lift-neg.f32 N/A

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

   13. remove-double-neg N/A

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

   14. frac-2neg N/A

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

   15. frac-2neg N/A

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

4. Applied egg-rr 99.2%

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

5. Final simplification 99.2%

   \[\leadsto \frac{e^{-\frac{\left|x\right|}{s}}}{s \cdot {\left(e^{-\frac{\left|x\right|}{s}} + 1\right)}^{2}} \]
6. Add Preprocessing

Alternative 6: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ {\left(t\_0 + 1\right)}^{-2} \cdot \frac{t\_0}{s} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s))))) (* (pow (+ t_0 1.0) -2.0) (/ t_0 s))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	return powf((t_0 + 1.0f), -2.0f) * (t_0 / s);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp(-(abs(x) / s))
    code = ((t_0 + 1.0e0) ** (-2.0e0)) * (t_0 / s)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	return Float32((Float32(t_0 + Float32(1.0)) ^ Float32(-2.0)) * Float32(t_0 / s))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp(-(abs(x) / s));
	tmp = ((t_0 + single(1.0)) ^ single(-2.0)) * (t_0 / s);
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-fabs.f32 N/A

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

   2. remove-double-neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. frac-2neg N/A

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

   6. frac-2neg N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-lft-identity N/A

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

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{{\left(e^{-\frac{\left|x\right|}{s}} + 1\right)}^{-2} \cdot \frac{e^{-\frac{\left|x\right|}{s}}}{s}} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{\left|x\right|}{s}\\ \frac{e^{\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{t\_0}\right), t\_0\right)}}{s} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (- (/ (fabs x) s))))
   (/ (exp (fma -2.0 (log1p (exp t_0)) t_0)) s)))

C

float code(float x, float s) {
	float t_0 = -(fabsf(x) / s);
	return expf(fmaf(-2.0f, log1pf(expf(t_0)), t_0)) / s;
}

Julia

function code(x, s)
	t_0 = Float32(-Float32(abs(x) / s))
	return Float32(exp(fma(Float32(-2.0), log1p(exp(t_0)), t_0)) / s)
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

4. Applied egg-rr 99.2%

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

   1. lift-fabs.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-+.f32 N/A

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

   6. lift-pow.f32 N/A

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

   7. lift-fabs.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift-*.f32 N/A

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

6. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right), \frac{\left|x\right|}{-s}\right)}}{s}} \]

7. Final simplification 99.2%

   \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \mathsf{log1p}\left(e^{-\frac{\left|x\right|}{s}}\right), -\frac{\left|x\right|}{s}\right)}}{s} \]
8. Add Preprocessing

Alternative 8: 97.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ \frac{e^{-t\_0} \cdot {\left(\frac{1}{\frac{\left|x\right| + \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(t\_0, 0.16666666666666666, 0.5\right)}{s}}{s} + 1} + 1\right)}^{-2}}{s} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) s)))
   (/
    (*
     (exp (- t_0))
     (pow
      (+
       (/
        1.0
        (+
         (/ (+ (fabs x) (/ (* (* x x) (fma t_0 0.16666666666666666 0.5)) s)) s)
         1.0))
       1.0)
      -2.0))
    s)))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / s;
	return (expf(-t_0) * powf(((1.0f / (((fabsf(x) + (((x * x) * fmaf(t_0, 0.16666666666666666f, 0.5f)) / s)) / s) + 1.0f)) + 1.0f), -2.0f)) / s;
}

Julia

function code(x, s)
	t_0 = Float32(abs(x) / s)
	return Float32(Float32(exp(Float32(-t_0)) * (Float32(Float32(Float32(1.0) / Float32(Float32(Float32(abs(x) + Float32(Float32(Float32(x * x) * fma(t_0, Float32(0.16666666666666666), Float32(0.5))) / s)) / s) + Float32(1.0))) + Float32(1.0)) ^ Float32(-2.0))) / s)
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

4. Applied egg-rr 99.2%

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

   1. lift-fabs.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lower-/.f32 N/A

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

   5. lower-exp.f32 99.2

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

6. Applied egg-rr 99.2%

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

7. Taylor expanded in s around -inf

   \[\leadsto \frac{{\left(\frac{1}{\color{blue}{1 + -1 \cdot \frac{-1 \cdot \left|x\right| + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}}{s}}{s}}} + 1\right)}^{-2} \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{s} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{{\left(\frac{1}{1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \left|x\right| + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}}{s}}{s}\right)\right)}} + 1\right)}^{-2} \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{s} \]

   2. unsub-neg N/A

      \[\leadsto \frac{{\left(\frac{1}{\color{blue}{1 - \frac{-1 \cdot \left|x\right| + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}}{s}}{s}}} + 1\right)}^{-2} \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{s} \]

   3. lower--.f32 N/A

      \[\leadsto \frac{{\left(\frac{1}{\color{blue}{1 - \frac{-1 \cdot \left|x\right| + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}}{s}}{s}}} + 1\right)}^{-2} \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{s} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{{\left(\frac{1}{1 - \color{blue}{\frac{-1 \cdot \left|x\right| + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{s} + \frac{1}{2} \cdot {\left(\left|x\right|\right)}^{2}}{s}}{s}}} + 1\right)}^{-2} \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{s} \]

9. Simplified 96.0%

   \[\leadsto \frac{{\left(\frac{1}{\color{blue}{1 - \frac{-\left(\left|x\right| + \frac{\mathsf{fma}\left(x \cdot x, 0.5, \frac{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right)}{s}\right)}{s}\right)}{s}}} + 1\right)}^{-2} \cdot e^{-\frac{\left|x\right|}{s}}}{s} \]

10. Taylor expanded in x around 0

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

    1. lower-*.f32 N/A

       \[\leadsto \frac{{\left(\frac{1}{1 - \frac{\mathsf{neg}\left(\left(\left|x\right| + \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \frac{\left|x\right|}{s}\right)}}{s}\right)\right)}{s}} + 1\right)}^{-2} \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{s} \]

    2. unpow2 N/A

       \[\leadsto \frac{{\left(\frac{1}{1 - \frac{\mathsf{neg}\left(\left(\left|x\right| + \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \frac{\left|x\right|}{s}\right)}{s}\right)\right)}{s}} + 1\right)}^{-2} \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{s} \]

    3. lower-*.f32 N/A

       \[\leadsto \frac{{\left(\frac{1}{1 - \frac{\mathsf{neg}\left(\left(\left|x\right| + \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \frac{\left|x\right|}{s}\right)}{s}\right)\right)}{s}} + 1\right)}^{-2} \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{s} \]

    4. +-commutative N/A

       \[\leadsto \frac{{\left(\frac{1}{1 - \frac{\mathsf{neg}\left(\left(\left|x\right| + \frac{\left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot \frac{\left|x\right|}{s} + \frac{1}{2}\right)}}{s}\right)\right)}{s}} + 1\right)}^{-2} \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{s} \]

    5. *-commutative N/A

       \[\leadsto \frac{{\left(\frac{1}{1 - \frac{\mathsf{neg}\left(\left(\left|x\right| + \frac{\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{\left|x\right|}{s} \cdot \frac{1}{6}} + \frac{1}{2}\right)}{s}\right)\right)}{s}} + 1\right)}^{-2} \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{s} \]

    6. lower-fma.f32 N/A

       \[\leadsto \frac{{\left(\frac{1}{1 - \frac{\mathsf{neg}\left(\left(\left|x\right| + \frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\left|x\right|}{s}, \frac{1}{6}, \frac{1}{2}\right)}}{s}\right)\right)}{s}} + 1\right)}^{-2} \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{s} \]

    7. lower-/.f32 N/A

       \[\leadsto \frac{{\left(\frac{1}{1 - \frac{\mathsf{neg}\left(\left(\left|x\right| + \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\left|x\right|}{s}}, \frac{1}{6}, \frac{1}{2}\right)}{s}\right)\right)}{s}} + 1\right)}^{-2} \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}}{s} \]

    8. lower-fabs.f32 96.1

       \[\leadsto \frac{{\left(\frac{1}{1 - \frac{-\left(\left|x\right| + \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{\left|x\right|}}{s}, 0.16666666666666666, 0.5\right)}{s}\right)}{s}} + 1\right)}^{-2} \cdot e^{-\frac{\left|x\right|}{s}}}{s} \]

12. Simplified 96.1%

    \[\leadsto \frac{{\left(\frac{1}{1 - \frac{-\left(\left|x\right| + \frac{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{\left|x\right|}{s}, 0.16666666666666666, 0.5\right)}}{s}\right)}{s}} + 1\right)}^{-2} \cdot e^{-\frac{\left|x\right|}{s}}}{s} \]

13. Final simplification 96.1%

    \[\leadsto \frac{e^{-\frac{\left|x\right|}{s}} \cdot {\left(\frac{1}{\frac{\left|x\right| + \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{\left|x\right|}{s}, 0.16666666666666666, 0.5\right)}{s}}{s} + 1} + 1\right)}^{-2}}{s} \]
14. Add Preprocessing

Alternative 9: 97.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{\left|x\right|}{s}} \cdot {\left(\frac{1}{1 - \frac{\mathsf{fma}\left(x \cdot x, \frac{-0.5}{s}, -\left|x\right|\right)}{s}} + 1\right)}^{-2}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  (*
   (exp (- (/ (fabs x) s)))
   (pow
    (+ (/ 1.0 (- 1.0 (/ (fma (* x x) (/ -0.5 s) (- (fabs x))) s))) 1.0)
    -2.0))
  s))

C

float code(float x, float s) {
	return (expf(-(fabsf(x) / s)) * powf(((1.0f / (1.0f - (fmaf((x * x), (-0.5f / s), -fabsf(x)) / s))) + 1.0f), -2.0f)) / s;
}

Julia

function code(x, s)
	return Float32(Float32(exp(Float32(-Float32(abs(x) / s))) * (Float32(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(fma(Float32(x * x), Float32(Float32(-0.5) / s), Float32(-abs(x))) / s))) + Float32(1.0)) ^ Float32(-2.0))) / s)
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

4. Applied egg-rr 99.2%

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

   1. lift-fabs.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lower-/.f32 N/A

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

   5. lower-exp.f32 99.2

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

6. Applied egg-rr 99.2%

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

7. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*l/ N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f32 N/A

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

   10. unpow2 N/A

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

   11. sqr-abs N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f32 N/A

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

   16. lower-fabs.f32 95.8

       \[\leadsto \frac{{\left(\frac{1}{1 - \frac{\mathsf{fma}\left(x \cdot x, \frac{-0.5}{s}, -\color{blue}{\left|x\right|}\right)}{s}} + 1\right)}^{-2} \cdot e^{-\frac{\left|x\right|}{s}}}{s} \]

9. Simplified 95.8%

   \[\leadsto \frac{{\left(\frac{1}{\color{blue}{1 - \frac{\mathsf{fma}\left(x \cdot x, \frac{-0.5}{s}, -\left|x\right|\right)}{s}}} + 1\right)}^{-2} \cdot e^{-\frac{\left|x\right|}{s}}}{s} \]

10. Final simplification 95.8%

    \[\leadsto \frac{e^{-\frac{\left|x\right|}{s}} \cdot {\left(\frac{1}{1 - \frac{\mathsf{fma}\left(x \cdot x, \frac{-0.5}{s}, -\left|x\right|\right)}{s}} + 1\right)}^{-2}}{s} \]
11. Add Preprocessing

Alternative 10: 96.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ \frac{e^{-t\_0} \cdot {\left(\frac{1}{t\_0 + 1} + 1\right)}^{-2}}{s} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) s)))
   (/ (* (exp (- t_0)) (pow (+ (/ 1.0 (+ t_0 1.0)) 1.0) -2.0)) s)))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / s;
	return (expf(-t_0) * powf(((1.0f / (t_0 + 1.0f)) + 1.0f), -2.0f)) / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = abs(x) / s
    code = (exp(-t_0) * (((1.0e0 / (t_0 + 1.0e0)) + 1.0e0) ** (-2.0e0))) / s
end function

Julia

function code(x, s)
	t_0 = Float32(abs(x) / s)
	return Float32(Float32(exp(Float32(-t_0)) * (Float32(Float32(Float32(1.0) / Float32(t_0 + Float32(1.0))) + Float32(1.0)) ^ Float32(-2.0))) / s)
end

MATLAB

function tmp = code(x, s)
	t_0 = abs(x) / s;
	tmp = (exp(-t_0) * (((single(1.0) / (t_0 + single(1.0))) + single(1.0)) ^ single(-2.0))) / s;
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

4. Applied egg-rr 99.2%

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

   1. lift-fabs.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. exp-neg N/A

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

   4. lower-/.f32 N/A

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

   5. lower-exp.f32 99.2

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

6. Applied egg-rr 99.2%

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

7. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-fabs.f32 95.5

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

9. Simplified 95.5%

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

10. Final simplification 95.5%

    \[\leadsto \frac{e^{-\frac{\left|x\right|}{s}} \cdot {\left(\frac{1}{\frac{\left|x\right|}{s} + 1} + 1\right)}^{-2}}{s} \]
11. Add Preprocessing

Alternative 11: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ \frac{1}{2 \cdot \left(e^{t\_0} \cdot \mathsf{fma}\left(s, e^{-t\_0}, s\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) s)))
   (/ 1.0 (* 2.0 (* (exp t_0) (fma s (exp (- t_0)) s))))))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / s;
	return 1.0f / (2.0f * (expf(t_0) * fmaf(s, expf(-t_0), s)));
}

Julia

function code(x, s)
	t_0 = Float32(abs(x) / s)
	return Float32(Float32(1.0) / Float32(Float32(2.0) * Float32(exp(t_0) * fma(s, exp(Float32(-t_0)), s))))
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in s around inf

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

5. Simplified 94.1%

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

7. Applied egg-rr 94.1%

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

8. Final simplification 94.1%

   \[\leadsto \frac{1}{2 \cdot \left(e^{\frac{\left|x\right|}{s}} \cdot \mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right)\right)} \]
9. Add Preprocessing

Alternative 12: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ \frac{t\_0}{2 \cdot \left(s \cdot \left(t\_0 + 1\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s))))) (/ t_0 (* 2.0 (* s (+ t_0 1.0))))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	return t_0 / (2.0f * (s * (t_0 + 1.0f)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp(-(abs(x) / s))
    code = t_0 / (2.0e0 * (s * (t_0 + 1.0e0)))
end function

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	return Float32(t_0 / Float32(Float32(2.0) * Float32(s * Float32(t_0 + Float32(1.0)))))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp(-(abs(x) / s));
	tmp = t_0 / (single(2.0) * (s * (t_0 + single(1.0))));
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in s around inf

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

5. Simplified 94.1%

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

6. Final simplification 94.1%

   \[\leadsto \frac{e^{-\frac{\left|x\right|}{s}}}{2 \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \]
7. Add Preprocessing

Alternative 13: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ \frac{t\_0 \cdot 0.5}{\mathsf{fma}\left(s, t\_0, s\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s))))) (/ (* t_0 0.5) (fma s t_0 s))))

C

float code(float x, float s) {
	float t_0 = expf(-(fabsf(x) / s));
	return (t_0 * 0.5f) / fmaf(s, t_0, s);
}

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	return Float32(Float32(t_0 * Float32(0.5)) / fma(s, t_0, s))
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in s around inf

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

5. Simplified 94.1%

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

   1. lift-fabs.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-exp.f32 N/A

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

7. Applied egg-rr 94.1%

   \[\leadsto \color{blue}{\frac{e^{\frac{\left|x\right|}{-s}} \cdot 0.5}{\mathsf{fma}\left(s, e^{\frac{\left|x\right|}{-s}}, s\right)}} \]

8. Final simplification 94.1%

   \[\leadsto \frac{e^{-\frac{\left|x\right|}{s}} \cdot 0.5}{\mathsf{fma}\left(s, e^{-\frac{\left|x\right|}{s}}, s\right)} \]
9. Add Preprocessing

Alternative 14: 94.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \left(e^{\frac{\left|x\right|}{s}} \cdot 4\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (* s (* (exp (/ (fabs x) s)) 4.0))))

C

float code(float x, float s) {
	return 1.0f / (s * (expf((fabsf(x) / s)) * 4.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * (exp((abs(x) / s)) * 4.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32(exp(Float32(abs(x) / s)) * Float32(4.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (s * (exp((abs(x) / s)) * single(4.0)));
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

4. Applied egg-rr 99.1%

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

5. Taylor expanded in s around inf

   \[\leadsto \frac{1}{s \cdot \left(\color{blue}{4} \cdot e^{\frac{\left|x\right|}{s}}\right)} \]
6. Step-by-step derivation

7. Simplified 93.7%

   \[\leadsto \frac{1}{s \cdot \left(\color{blue}{4} \cdot e^{\frac{\left|x\right|}{s}}\right)} \]

8. Final simplification 93.7%

   \[\leadsto \frac{1}{s \cdot \left(e^{\frac{\left|x\right|}{s}} \cdot 4\right)} \]
9. Add Preprocessing

Alternative 15: 94.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{\left|x\right|}{s}}}{s \cdot 4} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (exp (- (/ (fabs x) s))) (* s 4.0)))

C

float code(float x, float s) {
	return expf(-(fabsf(x) / s)) / (s * 4.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = exp(-(abs(x) / s)) / (s * 4.0e0)
end function

Julia

function code(x, s)
	return Float32(exp(Float32(-Float32(abs(x) / s))) / Float32(s * Float32(4.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = exp(-(abs(x) / s)) / (s * single(4.0));
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in s around inf

   \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{4 \cdot s}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}{\color{blue}{s \cdot 4}} \]

   2. lower-*.f32 93.7

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot 4}} \]

5. Simplified 93.7%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot 4}} \]

6. Final simplification 93.7%

   \[\leadsto \frac{e^{-\frac{\left|x\right|}{s}}}{s \cdot 4} \]
7. Add Preprocessing

Alternative 16: 82.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \mathbf{elif}\;\left|x\right| \leq 8.00000002901995 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{2 \cdot \frac{s \cdot \left(\frac{x \cdot x}{s \cdot s} - 4\right)}{\frac{\left|x\right|}{s} + 2}}\\ \mathbf{elif}\;\left|x\right| \leq 19999999655936:\\ \;\;\;\;\frac{-1}{s \cdot \left(\frac{\frac{\frac{\mathsf{fma}\left(\left|x\right|, \left(x \cdot x\right) \cdot 3, \left|x\right| \cdot \left(\left(x \cdot x\right) \cdot -3\right)\right)}{s} - x \cdot x}{s}}{s} - 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{\frac{x \cdot x}{s}}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 3.999999999279835e-23)
   (/ (fma (/ x s) (/ (* x -0.0625) s) 0.25) s)
   (if (<= (fabs x) 8.00000002901995e-15)
     (/
      -1.0
      (* 2.0 (/ (* s (- (/ (* x x) (* s s)) 4.0)) (+ (/ (fabs x) s) 2.0))))
     (if (<= (fabs x) 19999999655936.0)
       (/
        -1.0
        (*
         s
         (-
          (/
           (/
            (-
             (/ (fma (fabs x) (* (* x x) 3.0) (* (fabs x) (* (* x x) -3.0))) s)
             (* x x))
            s)
           s)
          4.0)))
       (/ 1.0 (* s (+ 4.0 (/ (/ (* x x) s) s))))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 3.999999999279835e-23f) {
		tmp = fmaf((x / s), ((x * -0.0625f) / s), 0.25f) / s;
	} else if (fabsf(x) <= 8.00000002901995e-15f) {
		tmp = -1.0f / (2.0f * ((s * (((x * x) / (s * s)) - 4.0f)) / ((fabsf(x) / s) + 2.0f)));
	} else if (fabsf(x) <= 19999999655936.0f) {
		tmp = -1.0f / (s * (((((fmaf(fabsf(x), ((x * x) * 3.0f), (fabsf(x) * ((x * x) * -3.0f))) / s) - (x * x)) / s) / s) - 4.0f));
	} else {
		tmp = 1.0f / (s * (4.0f + (((x * x) / s) / s)));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(3.999999999279835e-23))
		tmp = Float32(fma(Float32(x / s), Float32(Float32(x * Float32(-0.0625)) / s), Float32(0.25)) / s);
	elseif (abs(x) <= Float32(8.00000002901995e-15))
		tmp = Float32(Float32(-1.0) / Float32(Float32(2.0) * Float32(Float32(s * Float32(Float32(Float32(x * x) / Float32(s * s)) - Float32(4.0))) / Float32(Float32(abs(x) / s) + Float32(2.0)))));
	elseif (abs(x) <= Float32(19999999655936.0))
		tmp = Float32(Float32(-1.0) / Float32(s * Float32(Float32(Float32(Float32(Float32(fma(abs(x), Float32(Float32(x * x) * Float32(3.0)), Float32(abs(x) * Float32(Float32(x * x) * Float32(-3.0)))) / s) - Float32(x * x)) / s) / s) - Float32(4.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(Float32(Float32(x * x) / s) / s))));
	end
	return tmp
end

Derivation

1. Split input into 4 regimes

2. if (fabs.f32 x) < 4e-23

   1. Initial program 97.9%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]

   5. Simplified 54.6%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot s}}{s}} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{16}}{s \cdot s}}{s} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{16}}}{s \cdot s}}{s} \]

      3. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{\color{blue}{s \cdot s}}}{s} \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s}}}{s} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s} + \frac{1}{4}}}{s} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s}} + \frac{1}{4}}{s} \]

      7. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{16}}}{s \cdot s} + \frac{1}{4}}{s} \]

      8. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{16}}{s \cdot s} + \frac{1}{4}}{s} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \frac{-1}{16}\right)}}{s \cdot s} + \frac{1}{4}}{s} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{\frac{x \cdot \left(x \cdot \frac{-1}{16}\right)}{\color{blue}{s \cdot s}} + \frac{1}{4}}{s} \]

      11. times-frac N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{s} \cdot \frac{x \cdot \frac{-1}{16}}{s}} + \frac{1}{4}}{s} \]

      12. lower-fma.f32 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot \frac{-1}{16}}{s}, \frac{1}{4}\right)}}{s} \]

      13. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x}{s}}, \frac{x \cdot \frac{-1}{16}}{s}, \frac{1}{4}\right)}{s} \]

      14. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{x \cdot \frac{-1}{16}}{s}}, \frac{1}{4}\right)}{s} \]

      15. lower-*.f32 79.0

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \frac{\color{blue}{x \cdot -0.0625}}{s}, 0.25\right)}{s} \]

   7. Applied egg-rr 79.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}}{s} \]

   if 4e-23 < (fabs.f32 x) < 8.00000003e-15

   1. Initial program 98.1%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

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

   5. Simplified 93.9%

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

   6. Taylor expanded in s around inf

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

      1. lower-*.f32 N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fabs.f32 93.2

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

   8. Simplified 93.2%

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

   9. Taylor expanded in s around inf

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

   11. Simplified 30.2%

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

       1. lift-fabs.f32 N/A

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

       2. lift-/.f32 N/A

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

       3. lift--.f32 N/A

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

       4. *-commutative N/A

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

       5. lift--.f32 N/A

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

       6. flip-- N/A

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

       7. associate-*l/ N/A

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

       8. lower-/.f32 N/A

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

   13. Applied egg-rr 90.6%

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

   if 8.00000003e-15 < (fabs.f32 x) < 1.99999997e13

   1. Initial program 99.8%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in s around -inf

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

   7. Simplified 75.6%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{0 + \frac{\left(0 + x \cdot x\right) + \frac{0 + \mathsf{fma}\left(\left|x\right|, 3 \cdot \left(x \cdot x\right), \left|x\right| \cdot \left(-3 \cdot \left(x \cdot x\right)\right)\right)}{-s}}{-s}}{s}\right)}} \]

   if 1.99999997e13 < (fabs.f32 x)

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   7. Simplified 100.0%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{0 + \left(-\frac{0 + x \cdot x}{s}\right)}{s}\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \mathbf{elif}\;\left|x\right| \leq 8.00000002901995 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{2 \cdot \frac{s \cdot \left(\frac{x \cdot x}{s \cdot s} - 4\right)}{\frac{\left|x\right|}{s} + 2}}\\ \mathbf{elif}\;\left|x\right| \leq 19999999655936:\\ \;\;\;\;\frac{-1}{s \cdot \left(\frac{\frac{\frac{\mathsf{fma}\left(\left|x\right|, \left(x \cdot x\right) \cdot 3, \left|x\right| \cdot \left(\left(x \cdot x\right) \cdot -3\right)\right)}{s} - x \cdot x}{s}}{s} - 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{\frac{x \cdot x}{s}}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 80.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \mathbf{elif}\;\left|x\right| \leq 3.999999975690116 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{2 \cdot \frac{s \cdot \left(\frac{x \cdot x}{s \cdot s} - 4\right)}{\frac{\left|x\right|}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{\frac{x \cdot x}{s}}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 3.999999999279835e-23)
   (/ (fma (/ x s) (/ (* x -0.0625) s) 0.25) s)
   (if (<= (fabs x) 3.999999975690116e-8)
     (/
      -1.0
      (* 2.0 (/ (* s (- (/ (* x x) (* s s)) 4.0)) (+ (/ (fabs x) s) 2.0))))
     (/ 1.0 (* s (+ 4.0 (/ (/ (* x x) s) s)))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 3.999999999279835e-23f) {
		tmp = fmaf((x / s), ((x * -0.0625f) / s), 0.25f) / s;
	} else if (fabsf(x) <= 3.999999975690116e-8f) {
		tmp = -1.0f / (2.0f * ((s * (((x * x) / (s * s)) - 4.0f)) / ((fabsf(x) / s) + 2.0f)));
	} else {
		tmp = 1.0f / (s * (4.0f + (((x * x) / s) / s)));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(3.999999999279835e-23))
		tmp = Float32(fma(Float32(x / s), Float32(Float32(x * Float32(-0.0625)) / s), Float32(0.25)) / s);
	elseif (abs(x) <= Float32(3.999999975690116e-8))
		tmp = Float32(Float32(-1.0) / Float32(Float32(2.0) * Float32(Float32(s * Float32(Float32(Float32(x * x) / Float32(s * s)) - Float32(4.0))) / Float32(Float32(abs(x) / s) + Float32(2.0)))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(Float32(Float32(x * x) / s) / s))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (fabs.f32 x) < 4e-23

   1. Initial program 97.9%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]
   4. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]

   5. Simplified 54.6%

      \[\leadsto \color{blue}{\frac{0.25 + \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot s}}{s}} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{16}}{s \cdot s}}{s} \]

      2. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{16}}}{s \cdot s}}{s} \]

      3. lift-*.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{\color{blue}{s \cdot s}}}{s} \]

      4. lift-/.f32 N/A

         \[\leadsto \frac{\frac{1}{4} + \color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s}}}{s} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s} + \frac{1}{4}}}{s} \]

      6. lift-/.f32 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot \frac{-1}{16}}{s \cdot s}} + \frac{1}{4}}{s} \]

      7. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{16}}}{s \cdot s} + \frac{1}{4}}{s} \]

      8. lift-*.f32 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{16}}{s \cdot s} + \frac{1}{4}}{s} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \frac{-1}{16}\right)}}{s \cdot s} + \frac{1}{4}}{s} \]

      10. lift-*.f32 N/A

          \[\leadsto \frac{\frac{x \cdot \left(x \cdot \frac{-1}{16}\right)}{\color{blue}{s \cdot s}} + \frac{1}{4}}{s} \]

      11. times-frac N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{s} \cdot \frac{x \cdot \frac{-1}{16}}{s}} + \frac{1}{4}}{s} \]

      12. lower-fma.f32 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot \frac{-1}{16}}{s}, \frac{1}{4}\right)}}{s} \]

      13. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x}{s}}, \frac{x \cdot \frac{-1}{16}}{s}, \frac{1}{4}\right)}{s} \]

      14. lower-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{x \cdot \frac{-1}{16}}{s}}, \frac{1}{4}\right)}{s} \]

      15. lower-*.f32 79.0

          \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \frac{\color{blue}{x \cdot -0.0625}}{s}, 0.25\right)}{s} \]

   7. Applied egg-rr 79.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}}{s} \]

   if 4e-23 < (fabs.f32 x) < 3.99999998e-8

   1. Initial program 98.6%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in s around inf

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

   5. Simplified 92.2%

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

   6. Taylor expanded in s around inf

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

      1. lower-*.f32 N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fabs.f32 91.3

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

   8. Simplified 91.3%

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

   9. Taylor expanded in s around inf

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

   11. Simplified 28.1%

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

       1. lift-fabs.f32 N/A

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

       2. lift-/.f32 N/A

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

       3. lift--.f32 N/A

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

       4. *-commutative N/A

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

       5. lift--.f32 N/A

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

       6. flip-- N/A

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

       7. associate-*l/ N/A

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

       8. lower-/.f32 N/A

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

   13. Applied egg-rr 82.5%

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

   if 3.99999998e-8 < (fabs.f32 x)

   1. Initial program 99.9%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Add Preprocessing

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

   7. Simplified 79.7%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{0 + \left(-\frac{0 + x \cdot x}{s}\right)}{s}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \mathbf{elif}\;\left|x\right| \leq 3.999999975690116 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{2 \cdot \frac{s \cdot \left(\frac{x \cdot x}{s \cdot s} - 4\right)}{\frac{\left|x\right|}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{\frac{x \cdot x}{s}}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 75.7% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \left(4 + \frac{\frac{x \cdot x}{s}}{s}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (* s (+ 4.0 (/ (/ (* x x) s) s)))))

C

float code(float x, float s) {
	return 1.0f / (s * (4.0f + (((x * x) / s) / s)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * (4.0e0 + (((x * x) / s) / s)))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(Float32(Float32(x * x) / s) / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (s * (single(4.0) + (((x * x) / s) / s)));
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

4. Applied egg-rr 99.1%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Simplified 71.5%

   \[\leadsto \frac{1}{s \cdot \color{blue}{\left(4 - \frac{0 + \left(-\frac{0 + x \cdot x}{s}\right)}{s}\right)}} \]

8. Final simplification 71.5%

   \[\leadsto \frac{1}{s \cdot \left(4 + \frac{\frac{x \cdot x}{s}}{s}\right)} \]
9. Add Preprocessing

Alternative 19: 50.1% accurate, 9.1× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (* 2.0 (* s (fma (/ -1.0 s) (fabs x) 2.0)))))

C

float code(float x, float s) {
	return 1.0f / (2.0f * (s * fmaf((-1.0f / s), fabsf(x), 2.0f)));
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(2.0) * Float32(s * fma(Float32(Float32(-1.0) / s), abs(x), Float32(2.0)))))
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in s around inf

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

5. Simplified 94.1%

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

6. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

   2. neg-mul-1 N/A

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

   3. unsub-neg N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fabs.f32 93.5

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

8. Simplified 93.5%

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

9. Taylor expanded in s around inf

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

11. Simplified 47.1%

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

    1. lift-fabs.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. sub-neg N/A

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

    4. lift-/.f32 N/A

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

    5. distribute-frac-neg2 N/A

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

    6. lift-neg.f32 N/A

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

    7. lift-/.f32 N/A

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

    8. +-commutative N/A

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

    9. lift-/.f32 N/A

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

    10. clear-num N/A

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

    11. associate-/r/ N/A

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

    12. lower-fma.f32 N/A

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

    13. frac-2neg N/A

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

    14. metadata-eval N/A

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

    15. lift-neg.f32 N/A

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

    16. remove-double-neg N/A

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

    17. lower-/.f32 47.1

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

13. Applied egg-rr 47.1%

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

14. Final simplification 47.1%

    \[\leadsto \frac{1}{2 \cdot \left(s \cdot \mathsf{fma}\left(\frac{-1}{s}, \left|x\right|, 2\right)\right)} \]
15. Add Preprocessing

Alternative 20: 50.0% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{2 \cdot \left(s \cdot \left(\frac{\left|x\right|}{s} - 2\right)\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ -1.0 (* 2.0 (* s (- (/ (fabs x) s) 2.0)))))

C

float code(float x, float s) {
	return -1.0f / (2.0f * (s * ((fabsf(x) / s) - 2.0f)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (-1.0e0) / (2.0e0 * (s * ((abs(x) / s) - 2.0e0)))
end function

Julia

function code(x, s)
	return Float32(Float32(-1.0) / Float32(Float32(2.0) * Float32(s * Float32(Float32(abs(x) / s) - Float32(2.0)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(-1.0) / (single(2.0) * (s * ((abs(x) / s) - single(2.0))));
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in s around inf

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

5. Simplified 94.1%

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

6. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

   2. neg-mul-1 N/A

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

   3. unsub-neg N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fabs.f32 93.5

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

8. Simplified 93.5%

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

9. Taylor expanded in s around inf

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

11. Simplified 47.1%

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

12. Final simplification 47.1%

    \[\leadsto \frac{-1}{2 \cdot \left(s \cdot \left(\frac{\left|x\right|}{s} - 2\right)\right)} \]
13. Add Preprocessing

Alternative 21: 50.0% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s \cdot \left(2 - \frac{\left|x\right|}{s}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (* s (- 2.0 (/ (fabs x) s)))))

C

float code(float x, float s) {
	return 0.5f / (s * (2.0f - (fabsf(x) / s)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / (s * (2.0e0 - (abs(x) / s)))
end function

Julia

function code(x, s)
	return Float32(Float32(0.5) / Float32(s * Float32(Float32(2.0) - Float32(abs(x) / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / (s * (single(2.0) - (abs(x) / s)));
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in s around inf

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

5. Simplified 94.1%

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

6. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

   2. neg-mul-1 N/A

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

   3. unsub-neg N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fabs.f32 93.5

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

8. Simplified 93.5%

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

9. Taylor expanded in s around inf

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

11. Simplified 47.1%

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

    1. lift-fabs.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift--.f32 N/A

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

    4. lift-*.f32 N/A

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

    5. *-commutative N/A

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

    6. associate-/r* N/A

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

    7. metadata-eval N/A

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

    8. lower-/.f32 46.9

       \[\leadsto \color{blue}{\frac{0.5}{s \cdot \left(2 - \frac{\left|x\right|}{s}\right)}} \]

13. Applied egg-rr 46.9%

    \[\leadsto \color{blue}{\frac{0.5}{s \cdot \left(2 - \frac{\left|x\right|}{s}\right)}} \]
14. Add Preprocessing

Alternative 22: 26.6% accurate, 31.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 s))

C

float code(float x, float s) {
	return 0.25f / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / s
end function

Julia

function code(x, s)
	return Float32(Float32(0.25) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.25) / s;
end

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in s around inf

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

   1. lower-/.f32 28.6

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]

5. Simplified 28.6%

   \[\leadsto \color{blue}{\frac{0.25}{s}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (x s)
  :name "Logistic distribution"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ (exp (/ (- (fabs x)) s)) (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))