Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 14.4s

Alternatives: 14

Speedup: 1.5×

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, 1.5× speedup

Math

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

FPCore

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

C

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

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) / ((s + (s * exp(t_0))) * ((-1.0e0) - exp(-t_0)))
end function

Julia

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

MATLAB

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

Derivation

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
3. Step-by-step derivation

   1. associate-/l/ N/A

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

   2. /-lowering-/.f32 N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. exp-neg N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-rgt-identity N/A

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

   7. distribute-lft-in N/A

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

   8. associate-*r* N/A

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

7. Simplified 99.9%

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

   1. +-lowering-+.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. exp-lowering-exp.f32 N/A

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

   4. /-lowering-/.f32 N/A

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

   5. fabs-lowering-fabs.f32 99.9

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0.004999999888241291:\\ \;\;\;\;\frac{t\_0}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x \cdot -0.0625}{s}, \frac{x}{s}, 0.25\right)}{s}\\ \end{array} \end{array} \]

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))) 0.004999999888241291)
     (/ t_0 s)
     (/ (fma (/ (* x -0.0625) s) (/ x 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))) <= 0.004999999888241291f) {
		tmp = t_0 / s;
	} else {
		tmp = fmaf(((x * -0.0625f) / s), (x / 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(0.004999999888241291))
		tmp = Float32(t_0 / s);
	else
		tmp = Float32(fma(Float32(Float32(x * Float32(-0.0625)) / s), Float32(x / 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))))) < 0.00499999989

   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
   3. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. /-lowering-/.f32 N/A

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

   4. Applied egg-rr 99.9%

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

      1. div-inv N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-lft-in N/A

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

      4. times-frac N/A

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

      5. un-div-inv N/A

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

      6. pow2 N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

   6. Applied egg-rr 99.9%

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

   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. neg-lowering-neg.f32 N/A

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

      3. /-lowering-/.f32 N/A

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

      4. fabs-lowering-fabs.f32 99.9

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

   9. Simplified 99.9%

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

   if 0.00499999989 < (/.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 99.6%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.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 81.8%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. accelerator-lowering-fma.f32 N/A

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

      6. /-lowering-/.f32 N/A

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

      7. *-lowering-*.f32 N/A

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

      8. /-lowering-/.f32 93.9

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

   7. Applied egg-rr 93.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(e^{-\frac{\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \leq 0.004999999888241291:\\ \;\;\;\;\frac{e^{-\frac{\left|x\right|}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x \cdot -0.0625}{s}, \frac{x}{s}, 0.25\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0.004999999888241291:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(\left|x\right|, s, 0.5 \cdot \left(x \cdot x\right)\right)}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

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))) 0.004999999888241291)
     (/ 0.5 (/ (fma (fabs x) s (* 0.5 (* x x))) 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))) <= 0.004999999888241291f) {
		tmp = 0.5f / (fmaf(fabsf(x), s, (0.5f * (x * x))) / s);
	} else {
		tmp = 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(0.004999999888241291))
		tmp = Float32(Float32(0.5) / Float32(fma(abs(x), s, Float32(Float32(0.5) * Float32(x * x))) / s));
	else
		tmp = Float32(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))))) < 0.00499999989

   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
   3. Step-by-step derivation

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in s around inf

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

   7. Simplified 99.6%

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

   8. Taylor expanded in s around inf

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

      1. *-lowering-*.f32 N/A

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

      2. associate-+r+ N/A

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

      3. +-lowering-+.f32 N/A

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

      4. neg-mul-1 N/A

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

      5. unsub-neg N/A

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

      6. --lowering--.f32 N/A

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

      7. /-lowering-/.f32 N/A

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

      8. fabs-lowering-fabs.f32 N/A

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

   10. Simplified 1.9%

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

   11. Taylor expanded in s around 0

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

       1. /-lowering-/.f32 N/A

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

       2. +-commutative N/A

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

       3. associate-+l+ N/A

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

       4. *-commutative N/A

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

       5. sub-neg N/A

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

       6. mul-1-neg N/A

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

       7. distribute-rgt-out N/A

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

       8. metadata-eval N/A

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

       9. *-rgt-identity N/A

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

       10. +-commutative N/A

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

       11. accelerator-lowering-fma.f32 N/A

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

       12. fabs-lowering-fabs.f32 N/A

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

       13. distribute-lft1-in N/A

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

       14. metadata-eval N/A

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

       15. *-lowering-*.f32 N/A

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

       16. unpow2 N/A

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

       17. *-lowering-*.f32 57.4

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

   13. Simplified 57.4%

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

   if 0.00499999989 < (/.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 99.6%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 91.5

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

   5. Simplified 91.5%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(e^{-\frac{\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \leq 0.004999999888241291:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(\left|x\right|, s, 0.5 \cdot \left(x \cdot x\right)\right)}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0.004999999888241291:\\ \;\;\;\;\frac{0.5}{\frac{0.5 \cdot \left(x \cdot x\right)}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

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))) 0.004999999888241291)
     (/ 0.5 (/ (* 0.5 (* x x)) 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))) <= 0.004999999888241291f) {
		tmp = 0.5f / ((0.5f * (x * x)) / s);
	} 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))) <= 0.004999999888241291e0) then
        tmp = 0.5e0 / ((0.5e0 * (x * x)) / s)
    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(0.004999999888241291))
		tmp = Float32(Float32(0.5) / Float32(Float32(Float32(0.5) * Float32(x * x)) / s));
	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(0.004999999888241291))
		tmp = single(0.5) / ((single(0.5) * (x * x)) / s);
	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))))) < 0.00499999989

   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
   3. Step-by-step derivation

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in s around inf

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

   7. Simplified 99.6%

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

   8. Taylor expanded in s around inf

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

      1. *-lowering-*.f32 N/A

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

      2. associate-+r+ N/A

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

      3. +-lowering-+.f32 N/A

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

      4. neg-mul-1 N/A

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

      5. unsub-neg N/A

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

      6. --lowering--.f32 N/A

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

      7. /-lowering-/.f32 N/A

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

      8. fabs-lowering-fabs.f32 N/A

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

   10. Simplified 1.9%

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

   11. Taylor expanded in s around 0

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

       1. /-lowering-/.f32 N/A

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

       2. distribute-lft1-in N/A

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

       3. metadata-eval N/A

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

       4. *-lowering-*.f32 N/A

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

       5. unpow2 N/A

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

       6. *-lowering-*.f32 57.4

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

   13. Simplified 57.4%

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

   if 0.00499999989 < (/.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 99.6%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 91.5

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

   5. Simplified 91.5%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(e^{-\frac{\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \leq 0.004999999888241291:\\ \;\;\;\;\frac{0.5}{\frac{0.5 \cdot \left(x \cdot x\right)}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0.004999999888241291:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

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))) 0.004999999888241291)
     (/ s (* x x))
     (/ 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))) <= 0.004999999888241291f) {
		tmp = s / (x * x);
	} 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))) <= 0.004999999888241291e0) then
        tmp = s / (x * x)
    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(0.004999999888241291))
		tmp = Float32(s / Float32(x * x));
	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(0.004999999888241291))
		tmp = s / (x * x);
	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))))) < 0.00499999989

   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
   3. Step-by-step derivation

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in s around inf

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

   7. Simplified 99.6%

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

   8. Taylor expanded in s around inf

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

      1. *-lowering-*.f32 N/A

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

      2. associate-+r+ N/A

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

      3. +-lowering-+.f32 N/A

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

      4. neg-mul-1 N/A

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

      5. unsub-neg N/A

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

      6. --lowering--.f32 N/A

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

      7. /-lowering-/.f32 N/A

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

      8. fabs-lowering-fabs.f32 N/A

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

   10. Simplified 1.9%

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

   11. Taylor expanded in s around 0

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

       1. associate-*r/ N/A

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

       2. distribute-lft1-in N/A

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

       3. metadata-eval N/A

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

       4. times-frac N/A

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

       5. metadata-eval N/A

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

       6. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]

       7. /-lowering-/.f32 N/A

          \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]

       8. unpow2 N/A

          \[\leadsto \frac{s}{\color{blue}{x \cdot x}} \]

       9. *-lowering-*.f32 55.1

          \[\leadsto \frac{s}{\color{blue}{x \cdot x}} \]

   13. Simplified 55.1%

       \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]

   if 0.00499999989 < (/.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 99.6%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 91.5

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

   5. Simplified 91.5%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(e^{-\frac{\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \leq 0.004999999888241291:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 30.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\frac{\left|x\right|}{s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0.004999999888241291:\\ \;\;\;\;\frac{0.5}{\left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

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))) 0.004999999888241291)
     (/ 0.5 (fabs x))
     (/ 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))) <= 0.004999999888241291f) {
		tmp = 0.5f / fabsf(x);
	} 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))) <= 0.004999999888241291e0) then
        tmp = 0.5e0 / abs(x)
    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(0.004999999888241291))
		tmp = Float32(Float32(0.5) / abs(x));
	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(0.004999999888241291))
		tmp = single(0.5) / abs(x);
	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))))) < 0.00499999989

   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
   3. Step-by-step derivation

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in s around inf

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

   7. Simplified 99.6%

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

   8. Taylor expanded in s around inf

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

      1. distribute-lft-in N/A

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

      2. distribute-rgt-out N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-lft-in N/A

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

      6. *-lowering-*.f32 N/A

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

      7. +-commutative N/A

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

      8. +-lowering-+.f32 N/A

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

      9. /-lowering-/.f32 N/A

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

      10. fabs-lowering-fabs.f32 40.0

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

   10. Simplified 40.0%

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

   11. Taylor expanded in s around 0

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

       1. fabs-lowering-fabs.f32 9.5

          \[\leadsto \frac{0.5}{\color{blue}{\left|x\right|}} \]

   13. Simplified 9.5%

       \[\leadsto \frac{0.5}{\color{blue}{\left|x\right|}} \]

   if 0.00499999989 < (/.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 99.6%

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

   3. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 91.5

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

   5. Simplified 91.5%

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

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{-\frac{\left|x\right|}{s}}}{\left(e^{-\frac{\left|x\right|}{s}} + 1\right) \cdot \left(s \cdot \left(e^{-\frac{\left|x\right|}{s}} + 1\right)\right)} \leq 0.004999999888241291:\\ \;\;\;\;\frac{0.5}{\left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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
3. Step-by-step derivation

   1. associate-/l/ N/A

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

   2. /-lowering-/.f32 N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. exp-neg N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-rgt-identity N/A

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

   7. distribute-lft-in N/A

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

   8. associate-*r* N/A

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 8: 96.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = fabsf(x) / s;
	return (expf(-t_0) * powf((2.0f - t_0), -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) * ((2.0e0 - t_0) ** (-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(2.0) - t_0) ^ Float32(-2.0))) / s)
end

MATLAB

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

Derivation

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%

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

5. Taylor expanded in s around inf

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f32 N/A

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

   4. /-lowering-/.f32 N/A

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

   5. fabs-lowering-fabs.f32 97.3

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

7. Simplified 97.3%

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

8. Final simplification 97.3%

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

Alternative 9: 95.2% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (* 0.5 (/ 1.0 (fma s (exp (/ (fabs x) s)) s))))

C

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

Julia

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

Derivation

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

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 96.4%

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

6. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. neg-mul-1 N/A

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

   3. distribute-lft-in N/A

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

   4. neg-mul-1 N/A

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

   5. *-rgt-identity N/A

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

   6. associate-*r/ N/A

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

   7. *-lowering-*.f32 N/A

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

   8. exp-neg N/A

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

   9. associate-/l/ N/A

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

8. Simplified 96.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{1}{\mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]
9. Add Preprocessing

Alternative 10: 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 99.8%

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

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. *-lowering-*.f32 96.2

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

5. Simplified 96.2%

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

6. Final simplification 96.2%

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

Alternative 11: 85.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{s \cdot \left(\mathsf{fma}\left(x, \frac{x}{s \cdot s}, \frac{\left|x\right| \cdot 2}{s}\right) + \mathsf{fma}\left(\left|x\right|, \frac{-2}{s}, 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s \cdot \left(\frac{\left|x\right| - \frac{\mathsf{fma}\left(x \cdot x, -0.5, \frac{\mathsf{fma}\left(\left|x\right| \cdot 0.16666666666666666, x \cdot x, 0\right)}{-s}\right)}{s}}{s} - -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 4.999999969612645e-9)
   (/
    1.0
    (*
     s
     (+
      (fma x (/ x (* s s)) (/ (* (fabs x) 2.0) s))
      (fma (fabs x) (/ -2.0 s) 4.0))))
   (/
    0.5
    (*
     s
     (-
      (/
       (-
        (fabs x)
        (/
         (fma
          (* x x)
          -0.5
          (/ (fma (* (fabs x) 0.16666666666666666) (* x x) 0.0) (- s)))
         s))
       s)
      -2.0)))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 4.999999969612645e-9f) {
		tmp = 1.0f / (s * (fmaf(x, (x / (s * s)), ((fabsf(x) * 2.0f) / s)) + fmaf(fabsf(x), (-2.0f / s), 4.0f)));
	} else {
		tmp = 0.5f / (s * (((fabsf(x) - (fmaf((x * x), -0.5f, (fmaf((fabsf(x) * 0.16666666666666666f), (x * x), 0.0f) / -s)) / s)) / s) - -2.0f));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(4.999999969612645e-9))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(fma(x, Float32(x / Float32(s * s)), Float32(Float32(abs(x) * Float32(2.0)) / s)) + fma(abs(x), Float32(Float32(-2.0) / s), Float32(4.0)))));
	else
		tmp = Float32(Float32(0.5) / Float32(s * Float32(Float32(Float32(abs(x) - Float32(fma(Float32(x * x), Float32(-0.5), Float32(fma(Float32(abs(x) * Float32(0.16666666666666666)), Float32(x * x), Float32(0.0)) / Float32(-s))) / s)) / s) - Float32(-2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 4.99999997e-9

   1. Initial program 99.6%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f32 N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0

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

      1. exp-neg N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

   7. Simplified 99.6%

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

      1. +-lowering-+.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. exp-lowering-exp.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. fabs-lowering-fabs.f32 99.8

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in s around inf

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

   12. Simplified 78.6%

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

   if 4.99999997e-9 < (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
   3. Step-by-step derivation

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. flip3-+ N/A

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

      6. clear-num N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in s around inf

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

   7. Simplified 100.0%

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

   8. Taylor expanded in s around -inf

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

   10. Simplified 87.8%

       \[\leadsto \frac{0.5}{\color{blue}{-s \cdot \left(-2 + \frac{\left|x\right| + \frac{\mathsf{fma}\left(x \cdot x, -0.5, \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left|x\right|, x \cdot x, 0\right)}{-s}\right)}{-s}}{-s}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{s \cdot \left(\mathsf{fma}\left(x, \frac{x}{s \cdot s}, \frac{\left|x\right| \cdot 2}{s}\right) + \mathsf{fma}\left(\left|x\right|, \frac{-2}{s}, 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s \cdot \left(\frac{\left|x\right| - \frac{\mathsf{fma}\left(x \cdot x, -0.5, \frac{\mathsf{fma}\left(\left|x\right| \cdot 0.16666666666666666, x \cdot x, 0\right)}{-s}\right)}{s}}{s} - -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.2500000392179937 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x \cdot -0.0625}{s}, \frac{x}{s}, 0.25\right)}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{s \cdot \left(-4 - \frac{x \cdot x}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 1.2500000392179937e-23)
   (/ (fma (/ (* x -0.0625) s) (/ x s) 0.25) s)
   (/ -1.0 (* s (- -4.0 (/ (* x x) (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 1.2500000392179937e-23f) {
		tmp = fmaf(((x * -0.0625f) / s), (x / s), 0.25f) / s;
	} 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(1.2500000392179937e-23))
		tmp = Float32(fma(Float32(Float32(x * Float32(-0.0625)) / s), Float32(x / s), Float32(0.25)) / s);
	else
		tmp = Float32(Float32(-1.0) / Float32(s * Float32(Float32(-4.0) - Float32(Float32(x * x) / Float32(s * s)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 1.25000004e-23

   1. Initial program 99.7%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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. /-lowering-/.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 65.2%

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. accelerator-lowering-fma.f32 N/A

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

      6. /-lowering-/.f32 N/A

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

      7. *-lowering-*.f32 N/A

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

      8. /-lowering-/.f32 82.7

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

   7. Applied egg-rr 82.7%

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

   if 1.25000004e-23 < (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
   3. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. /-lowering-/.f32 N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. exp-neg N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*r* N/A

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

   7. Simplified 99.9%

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

      1. +-lowering-+.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. exp-lowering-exp.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. fabs-lowering-fabs.f32 99.9

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in s around -inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f32 N/A

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

   12. Simplified 80.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.2500000392179937 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x \cdot -0.0625}{s}, \frac{x}{s}, 0.25\right)}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{s \cdot \left(-4 - \frac{x \cdot x}{s \cdot s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.3% accurate, 9.1× speedup

Math

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

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(x * x) / Float32(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 99.8%

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

   1. associate-/l/ N/A

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

   2. /-lowering-/.f32 N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. exp-neg N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-rgt-identity N/A

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

   7. distribute-lft-in N/A

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

   8. associate-*r* N/A

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

7. Simplified 99.9%

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

   1. +-lowering-+.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. exp-lowering-exp.f32 N/A

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

   4. /-lowering-/.f32 N/A

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

   5. fabs-lowering-fabs.f32 99.9

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

9. Applied egg-rr 99.9%

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

10. Taylor expanded in s around -inf

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. *-lowering-*.f32 N/A

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

12. Simplified 77.6%

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

13. Final simplification 77.6%

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

Alternative 14: 27.3% 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 99.8%

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

3. Taylor expanded in s around inf

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

   1. /-lowering-/.f32 27.3

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

5. Simplified 27.3%

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

Reproduce

herbie shell --seed 2024199 
(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))))))