Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 11.7s

Alternatives: 13

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-/.f32 N/A

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

   6. neg-mul-1 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-exp.f32 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0:\\ \;\;\;\;\frac{t\_0}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x}{s}, s \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x) (- s)))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 0.0)
     (/ t_0 s)
     (/ 1.0 (fma x (/ x s) (* s 4.0))))))

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 0.0f) {
		tmp = t_0 / s;
	} else {
		tmp = 1.0f / fmaf(x, (x / s), (s * 4.0f));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(0.0))
		tmp = Float32(t_0 / s);
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(x / s), Float32(s * Float32(4.0))));
	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))))) < 0.0

   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. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lift-*.f32 N/A

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

      6. associate-/l/ N/A

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

      7. lift-exp.f32 N/A

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

      8. exp-neg N/A

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

      9. lift-neg.f32 N/A

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

      10. lift-exp.f32 N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in s around 0

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

      1. neg-mul-1 N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-fabs.f32 100.0

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

   9. Applied rewrites 100.0%

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

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

   1. Initial program 98.8%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in s around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 N/A

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

   7. Applied rewrites 86.5%

      \[\leadsto \frac{1}{\color{blue}{\left(-4 - \frac{\mathsf{fma}\left(-4, \frac{x \cdot x}{s}, \frac{\left(x \cdot x\right) \cdot 5}{s}\right)}{s}\right) \cdot \left(-s\right)}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 88.5%

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

4. Final simplification 97.0%

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

Alternative 3: 85.1% accurate, 0.9× speedup

Math

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

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))) 4.0)
     (/ 1.0 (* s (fma x (/ x (* s s)) 4.0)))
     (/ 1.0 (fma x (/ x s) (* s 4.0))))))

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))) <= 4.0f) {
		tmp = 1.0f / (s * fmaf(x, (x / (s * s)), 4.0f));
	} else {
		tmp = 1.0f / fmaf(x, (x / s), (s * 4.0f));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-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(4.0))
		tmp = Float32(Float32(1.0) / Float32(s * fma(x, Float32(x / Float32(s * s)), Float32(4.0))));
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(x / s), Float32(s * Float32(4.0))));
	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))))) < 4

   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. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 N/A

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

   7. Applied rewrites 24.3%

      \[\leadsto \frac{1}{\color{blue}{\left(-4 - \frac{\mathsf{fma}\left(-4, \frac{x \cdot x}{s}, \frac{\left(x \cdot x\right) \cdot 5}{s}\right)}{s}\right) \cdot \left(-s\right)}} \]

   8. Taylor expanded in s around inf

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

   10. Applied rewrites 87.0%

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

   if 4 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 98.8%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in s around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 N/A

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

   7. Applied rewrites 86.1%

      \[\leadsto \frac{1}{\color{blue}{\left(-4 - \frac{\mathsf{fma}\left(-4, \frac{x \cdot x}{s}, \frac{\left(x \cdot x\right) \cdot 5}{s}\right)}{s}\right) \cdot \left(-s\right)}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 88.3%

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

4. Final simplification 87.3%

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

Alternative 4: 64.4% accurate, 0.9× speedup

Math

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

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))) 0.0) (/ 1.0 (/ (* 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))) <= 0.0f) {
		tmp = 1.0f / ((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))) <= 0.0e0) then
        tmp = 1.0e0 / ((x * x) / s)
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(0.0))
		tmp = Float32(Float32(1.0) / Float32(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(0.0))
		tmp = single(1.0) / ((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))))) < 0.0

   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. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f32 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 N/A

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

   7. Applied rewrites 22.5%

      \[\leadsto \frac{1}{\color{blue}{\left(-4 - \frac{\mathsf{fma}\left(-4, \frac{x \cdot x}{s}, \frac{\left(x \cdot x\right) \cdot 5}{s}\right)}{s}\right) \cdot \left(-s\right)}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 57.2%

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

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

   1. Initial program 98.8%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 84.2

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

   5. Applied rewrites 84.2%

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

4. Final simplification 64.2%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/l/ N/A

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

   7. lift-exp.f32 N/A

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

   8. exp-neg N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-exp.f32 N/A

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 7: 97.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. associate-*r/ N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. associate-*r/ N/A

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

   10. div-sub N/A

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

   11. unsub-neg N/A

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

   12. mul-1-neg N/A

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

   13. +-commutative N/A

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

5. Applied rewrites 95.8%

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

7. Applied rewrites 96.4%

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

8. Final simplification 96.4%

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

Alternative 8: 96.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. +-commutative N/A

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

   3. neg-mul-1 N/A

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

   4. unsub-neg N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. associate-*r/ N/A

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

   9. unpow2 N/A

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

   10. associate-/r* N/A

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

   11. associate-*r/ N/A

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

   12. div-sub N/A

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

   13. unsub-neg N/A

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

   14. mul-1-neg N/A

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

   15. +-commutative N/A

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

   16. lower-/.f32 N/A

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

7. Applied rewrites 96.2%

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

8. Final simplification 96.2%

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

Alternative 9: 96.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. *-lft-identity N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. times-frac N/A

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

   9. lower-*.f32 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. +-commutative N/A

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

   3. neg-mul-1 N/A

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

   4. unsub-neg N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. associate-*r/ N/A

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

   9. unpow2 N/A

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

   10. associate-/r* N/A

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

   11. associate-*r/ N/A

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

   12. div-sub N/A

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

   13. unsub-neg N/A

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

   14. mul-1-neg N/A

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

   15. +-commutative N/A

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

   16. lower-/.f32 N/A

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

7. Applied rewrites 96.2%

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

8. Final simplification 96.2%

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

Alternative 10: 96.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{\left|x\right|}{-s}} \cdot \frac{1}{s}}{4 - \frac{\mathsf{fma}\left(\left|x\right|, 4, \frac{\left(x \cdot x\right) \cdot -3}{s}\right)}{s}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  (* (exp (/ (fabs x) (- s))) (/ 1.0 s))
  (- 4.0 (/ (fma (fabs x) 4.0 (/ (* (* x x) -3.0) s)) s))))

C

float code(float x, float s) {
	return (expf((fabsf(x) / -s)) * (1.0f / s)) / (4.0f - (fmaf(fabsf(x), 4.0f, (((x * x) * -3.0f) / s)) / s));
}

Julia

function code(x, s)
	return Float32(Float32(exp(Float32(abs(x) / Float32(-s))) * Float32(Float32(1.0) / s)) / Float32(Float32(4.0) - Float32(fma(abs(x), Float32(4.0), Float32(Float32(Float32(x * x) * Float32(-3.0)) / s)) / s)))
end

Derivation

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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

   8. associate-*l/ N/A

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 95.2%

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

8. Final simplification 95.2%

   \[\leadsto \frac{e^{\frac{\left|x\right|}{-s}} \cdot \frac{1}{s}}{4 - \frac{\mathsf{fma}\left(\left|x\right|, 4, \frac{\left(x \cdot x\right) \cdot -3}{s}\right)}{s}} \]
9. Add Preprocessing

Alternative 11: 94.7% 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(abs(x) / Float32(-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 99.7%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 94.0

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

5. Applied rewrites 94.0%

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

6. Final simplification 94.0%

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

Alternative 12: 65.4% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in s around -inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

7. Applied rewrites 39.2%

   \[\leadsto \frac{1}{\color{blue}{\left(-4 - \frac{\mathsf{fma}\left(-4, \frac{x \cdot x}{s}, \frac{\left(x \cdot x\right) \cdot 5}{s}\right)}{s}\right) \cdot \left(-s\right)}} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 65.4%

    \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{x}{s}}, s \cdot 4\right)} \]
11. Add Preprocessing

Alternative 13: 27.2% 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 99.7%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 25.3

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

5. Applied rewrites 25.3%

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

Reproduce

herbie shell --seed 2024237 
(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))))))