Logistic distribution

Percentage Accurate: 99.6% → 99.3%

Time: 10.4s

Alternatives: 14

Speedup: N/A×

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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

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

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. distribute-frac-neg N/A

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

   6. exp-neg N/A

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

   7. associate-/l/ N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-exp.f32 N/A

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

   2. exp-1-e N/A

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

   3. lower-E.f32 99.8

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

8. Applied rewrites 99.8%

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

Alternative 2: 43.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(4, s, \frac{x \cdot x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* s t_1) t_1)) 4.99999991225835e-14)
     (/ 1.0 (fma 4.0 s (/ (* x x) s)))
     (/ 0.25 s))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	float tmp;
	if ((t_0 / ((s * t_1) * t_1)) <= 4.99999991225835e-14f) {
		tmp = 1.0f / fmaf(4.0f, s, ((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(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(s * t_1) * t_1)) <= Float32(4.99999991225835e-14))
		tmp = Float32(Float32(1.0) / fma(Float32(4.0), s, Float32(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))))) < 4.99999991e-14

   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. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 99.9%

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

   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. lower-*.f32 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

   7. Applied rewrites 68.9%

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

   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 49.0%

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

   if 4.99999991e-14 < (/.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 99.2%

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

   3. Taylor expanded in s around inf

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

      1. lower-/.f32 87.6

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

   5. Applied rewrites 87.6%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. distribute-frac-neg N/A

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

   6. exp-neg N/A

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

   7. associate-/l/ N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 99.8%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

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

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. distribute-frac-neg N/A

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

   6. exp-neg N/A

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

   7. associate-/l/ N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-exp.f32 N/A

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

   2. exp-1-e N/A

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

   3. lower-E.f32 99.8

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

8. Applied rewrites 99.8%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f32 99.7

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

   5. lift-+.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f32 99.7

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

10. Applied rewrites 99.7%

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

Alternative 5: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return (powf((1.0f + t_0), -2.0f) * (1.0f / s)) * 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 = exp((-abs(x) / s))
    code = (((1.0e0 + t_0) ** (-2.0e0)) * (1.0e0 / s)) * t_0
end function

Julia

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

MATLAB

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

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

4. Applied rewrites 99.7%

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

Alternative 6: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return (powf((t_0 + 1.0f), -2.0f) / s) * 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 = exp((-abs(x) / s))
    code = (((t_0 + 1.0e0) ** (-2.0e0)) / s) * t_0
end function

Julia

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

MATLAB

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

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. distribute-frac-neg N/A

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

   6. exp-neg N/A

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

   7. associate-/l/ N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-pow.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. pow-exp N/A

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

   6. rec-exp N/A

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

   7. *-lft-identity N/A

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

   8. lift-/.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. lift-neg.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. lower-*.f32 99.7

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

8. Applied rewrites 99.7%

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

Alternative 7: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. distribute-frac-neg N/A

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

   6. exp-neg N/A

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

   7. associate-/l/ N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. un-div-inv N/A

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

   5. associate-/l/ N/A

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

   6. lift-pow.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. pow-exp N/A

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

   9. *-lft-identity N/A

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

   10. lift-exp.f32 N/A

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

   11. lower-/.f32 N/A

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

8. Applied rewrites 99.7%

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

Alternative 8: 96.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / ((((((((x * x) / s) * 0.5f) - fabsf(x)) / s) + 2.0f) * s) * (1.0f + 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 = exp((-abs(x) / s))
    code = t_0 / ((((((((x * x) / s) * 0.5e0) - abs(x)) / s) + 2.0e0) * s) * (1.0e0 + t_0))
end function

Julia

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

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	tmp = t_0 / ((((((((x * x) / s) * single(0.5)) - abs(x)) / s) + single(2.0)) * s) * (single(1.0) + t_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

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 95.9%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(\left(\frac{\frac{x \cdot x}{s} \cdot 0.5 - \left|x\right|}{s} + 2\right) \cdot s\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
6. Add Preprocessing

Alternative 9: 96.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 95.6

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

7. Applied rewrites 95.6%

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

Alternative 10: 94.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot \frac{1}{s}}{e^{\frac{\left|x\right|}{s}}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (* 0.25 (/ 1.0 s)) (exp (/ (fabs x) s))))

C

float code(float x, float s) {
	return (0.25f * (1.0f / s)) / expf((fabsf(x) / s));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (0.25e0 * (1.0e0 / s)) / exp((abs(x) / s))
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(0.25) * Float32(Float32(1.0) / s)) / exp(Float32(abs(x) / s)))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(0.25) * (single(1.0) / s)) / 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

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. distribute-frac-neg N/A

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

   6. exp-neg N/A

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

   7. associate-/l/ N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 93.6%

   \[\leadsto \frac{\color{blue}{0.25} \cdot \frac{1}{s}}{e^{\frac{\left|x\right|}{s}}} \]
8. Add Preprocessing

Alternative 11: 94.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = exp((-abs(x) / s)) / (single(4.0) * 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. Taylor expanded in s around inf

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

   1. lower-*.f32 93.6

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

5. Applied rewrites 93.6%

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

Alternative 12: 82.5% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 3.999999999279835e-23) then
        tmp = (((((-0.0625e0) * x) * (x / s)) / s) + 0.25e0) / s
    else
        tmp = 1.0e0 / (s * (((x * x) / (s * s)) + 4.0e0))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(3.999999999279835e-23))
		tmp = ((((single(-0.0625) * x) * (x / s)) / s) + single(0.25)) / s;
	else
		tmp = single(1.0) / (s * (((x * x) / (s * s)) + single(4.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 4e-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. lower-/.f32 N/A

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 77.8%

      \[\leadsto \frac{\frac{\left(-0.0625 \cdot x\right) \cdot \frac{x}{s}}{s} + 0.25}{s} \]

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

   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

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 99.7%

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

   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. lower-*.f32 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{-1 \cdot \color{blue}{\left(s \cdot \left(\left(-5 \cdot \frac{{x}^{2}}{{s}^{2}} + 4 \cdot \frac{{x}^{2}}{{s}^{2}}\right) - 4\right)\right)}} \]

   10. Applied rewrites 79.8%

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

4. Final simplification 79.4%

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

Alternative 13: 82.2% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 3.999999999279835e-23) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (s * (((x * x) / (s * s)) + 4.0e0))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(3.999999999279835e-23))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (s * (((x * x) / (s * s)) + single(4.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

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

      1. lower-/.f32 76.9

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

   5. Applied rewrites 76.9%

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

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

   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

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

      2. clear-num N/A

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

      3. lower-/.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 99.7%

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

   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. lower-*.f32 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{-1 \cdot \color{blue}{\left(s \cdot \left(\left(-5 \cdot \frac{{x}^{2}}{{s}^{2}} + 4 \cdot \frac{{x}^{2}}{{s}^{2}}\right) - 4\right)\right)}} \]

   10. Applied rewrites 79.8%

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

4. Final simplification 79.1%

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

Alternative 14: 26.5% 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 31.1

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

5. Applied rewrites 31.1%

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

Reproduce

herbie shell --seed 2024325 
(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))))))