Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 12.3s

Alternatives: 12

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: 99.5% 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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. 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

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f32 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ t_1 := e^{\frac{-\left|x\right|}{s}}\\ t_2 := t\_0 + 1\\ t_3 := 1 + t\_1\\ \mathbf{if}\;\frac{t\_1}{\left(t\_3 \cdot s\right) \cdot t\_3} \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{e^{-t\_0}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\mathsf{fma}\left(\frac{x}{s}, 5 \cdot \frac{x}{s}, \mathsf{fma}\left(t\_2, 4, -4 \cdot \left(t\_2 \cdot t\_0\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) s))
        (t_1 (exp (/ (- (fabs x)) s)))
        (t_2 (+ t_0 1.0))
        (t_3 (+ 1.0 t_1)))
   (if (<= (/ t_1 (* (* t_3 s) t_3)) 9.99999993922529e-9)
     (/ (exp (- t_0)) s)
     (/
      (/ 1.0 s)
      (fma (/ x s) (* 5.0 (/ x s)) (fma t_2 4.0 (* -4.0 (* t_2 t_0))))))))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / s;
	float t_1 = expf((-fabsf(x) / s));
	float t_2 = t_0 + 1.0f;
	float t_3 = 1.0f + t_1;
	float tmp;
	if ((t_1 / ((t_3 * s) * t_3)) <= 9.99999993922529e-9f) {
		tmp = expf(-t_0) / s;
	} else {
		tmp = (1.0f / s) / fmaf((x / s), (5.0f * (x / s)), fmaf(t_2, 4.0f, (-4.0f * (t_2 * t_0))));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(abs(x) / s)
	t_1 = exp(Float32(Float32(-abs(x)) / s))
	t_2 = Float32(t_0 + Float32(1.0))
	t_3 = Float32(Float32(1.0) + t_1)
	tmp = Float32(0.0)
	if (Float32(t_1 / Float32(Float32(t_3 * s) * t_3)) <= Float32(9.99999993922529e-9))
		tmp = Float32(exp(Float32(-t_0)) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) / fma(Float32(x / s), Float32(Float32(5.0) * Float32(x / s)), fma(t_2, Float32(4.0), Float32(Float32(-4.0) * Float32(t_2 * t_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))))) < 9.99999994e-9

   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

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f32 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lift-+.f32 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f32 N/A

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

      11. associate-/l/ N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. lower-fabs.f32 99.9

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

   9. Applied rewrites 99.9%

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

   if 9.99999994e-9 < (/.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 99.3%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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. lift-*.f32 N/A

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

      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. associate-*l* N/A

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

      6. associate-/l* N/A

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

      7. associate-/r* N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. distribute-lft-out N/A

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

      5. lower-fma.f32 N/A

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

   7. Applied rewrites 79.6%

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

   9. Applied rewrites 90.5%

      \[\leadsto \frac{\frac{1}{s}}{\mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{x}{s} \cdot 5}, \mathsf{fma}\left(\frac{\left|x\right|}{s} + 1, 4, \left(\frac{\left|x\right|}{s} \cdot \left(\frac{\left|x\right|}{s} + 1\right)\right) \cdot -4\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot s\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{e^{-\frac{\left|x\right|}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\mathsf{fma}\left(\frac{x}{s}, 5 \cdot \frac{x}{s}, \mathsf{fma}\left(\frac{\left|x\right|}{s} + 1, 4, -4 \cdot \left(\left(\frac{\left|x\right|}{s} + 1\right) \cdot \frac{\left|x\right|}{s}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s \cdot s} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.0625 \cdot x}{s}, \frac{x}{s}, 0.25\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 9.99999993922529e-9)
     (/ (/ 1.0 s) (* (/ x (* s s)) x))
     (/ (fma (/ (* -0.0625 x) s) (/ 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 / ((t_1 * s) * t_1)) <= 9.99999993922529e-9f) {
		tmp = (1.0f / s) / ((x / (s * s)) * x);
	} else {
		tmp = fmaf(((-0.0625f * x) / s), (x / s), 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(t_1 * s) * t_1)) <= Float32(9.99999993922529e-9))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(x / Float32(s * s)) * x));
	else
		tmp = Float32(fma(Float32(Float32(Float32(-0.0625) * x) / s), Float32(x / s), Float32(0.25)) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 9.99999994e-9

   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

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

      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. lift-*.f32 N/A

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

      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. associate-*l* N/A

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

      6. associate-/l* N/A

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

      7. associate-/r* N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. distribute-lft-out N/A

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

      5. lower-fma.f32 N/A

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 82.3%

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

   if 9.99999994e-9 < (/.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 99.3%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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. lift-*.f32 N/A

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

      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. associate-*l* N/A

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

      6. associate-/l* N/A

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

      7. associate-/r* N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + -1 \cdot \frac{\frac{-1}{64} \cdot {\left(-4 \cdot \left|x\right| + 4 \cdot \left|x\right|\right)}^{2} + \frac{1}{16} \cdot \left(-4 \cdot {\left(\left|x\right|\right)}^{2} + \left(4 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{-4 \cdot \left|x\right| + 4 \cdot \left|x\right|}{s}}{s}} \]

   7. Applied rewrites 78.1%

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

   9. Applied rewrites 88.5%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot s\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s \cdot s} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.0625 \cdot x}{s}, \frac{x}{s}, 0.25\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \mathbf{if}\;\frac{t\_0}{\left(t\_1 \cdot s\right) \cdot t\_1} \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s \cdot s} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (if (<= (/ t_0 (* (* t_1 s) t_1)) 9.99999993922529e-9)
     (/ (/ 1.0 s) (* (/ x (* s s)) x))
     (/ 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 / ((t_1 * s) * t_1)) <= 9.99999993922529e-9f) {
		tmp = (1.0f / s) / ((x / (s * s)) * x);
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    if ((t_0 / ((t_1 * s) * t_1)) <= 9.99999993922529e-9) then
        tmp = (1.0e0 / s) / ((x / (s * s)) * x)
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(Float32(t_1 * s) * t_1)) <= Float32(9.99999993922529e-9))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(x / Float32(s * s)) * x));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = single(0.0);
	if ((t_0 / ((t_1 * s) * t_1)) <= single(9.99999993922529e-9))
		tmp = (single(1.0) / s) / ((x / (s * s)) * x);
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 9.99999994e-9

   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

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

      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. lift-*.f32 N/A

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

      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. associate-*l* N/A

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

      6. associate-/l* N/A

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

      7. associate-/r* N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. distribute-lft-out N/A

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

      5. lower-fma.f32 N/A

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 82.3%

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

   if 9.99999994e-9 < (/.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 99.3%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 87.7

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

   5. Applied rewrites 87.7%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{-\left|x\right|}{s}}}{\left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot s\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s \cdot s} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. 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. lift-*.f32 N/A

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

   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. associate-*l* N/A

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

   6. associate-/l* N/A

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

   7. associate-/r* N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-exp.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-fabs.f32 N/A

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

   9. lower-pow.f32 N/A

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

7. Applied rewrites 99.8%

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

8. Final simplification 99.8%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

Alternative 7: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f32 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. distribute-lft1-in N/A

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

   5. +-commutative N/A

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

   6. lift-+.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f32 N/A

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

   11. associate-/l/ N/A

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

6. Applied rewrites 99.8%

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

Alternative 8: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-\mathsf{fma}\left(\log 2 - \frac{\mathsf{fma}\left(x, -0.125 \cdot \frac{x}{s}, 0.5 \cdot \left|x\right|\right)}{s}, 2, \frac{\left|x\right|}{s}\right)}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  (exp
   (-
    (fma
     (- (log 2.0) (/ (fma x (* -0.125 (/ x s)) (* 0.5 (fabs x))) s))
     2.0
     (/ (fabs x) s))))
  s))

C

float code(float x, float s) {
	return expf(-fmaf((logf(2.0f) - (fmaf(x, (-0.125f * (x / s)), (0.5f * fabsf(x))) / s)), 2.0f, (fabsf(x) / s))) / s;
}

Julia

function code(x, s)
	return Float32(exp(Float32(-fma(Float32(log(Float32(2.0)) - Float32(fma(x, Float32(Float32(-0.125) * Float32(x / s)), Float32(Float32(0.5) * abs(x))) / s)), Float32(2.0), Float32(abs(x) / s)))) / s)
end

Derivation

1. Initial program 99.8%

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

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f32 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. distribute-lft1-in N/A

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

   5. +-commutative N/A

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

   6. lift-+.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f32 N/A

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

   11. associate-/l/ N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. lower-/.f32 N/A

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

9. Applied rewrites 95.2%

   \[\leadsto \frac{e^{-\mathsf{fma}\left(\color{blue}{\log 2 - \frac{\mathsf{fma}\left(\frac{x \cdot x}{s} \cdot 0.25, -0.5, 0.5 \cdot \left|x\right|\right)}{s}}, 2, \frac{\left|x\right|}{s}\right)}}{s} \]
10. Step-by-step derivation

11. Applied rewrites 97.1%

    \[\leadsto \frac{e^{-\mathsf{fma}\left(\log 2 - \frac{\mathsf{fma}\left(x, \frac{x}{s} \cdot -0.125, 0.5 \cdot \left|x\right|\right)}{s}, 2, \frac{\left|x\right|}{s}\right)}}{s} \]

12. Final simplification 97.1%

    \[\leadsto \frac{e^{-\mathsf{fma}\left(\log 2 - \frac{\mathsf{fma}\left(x, -0.125 \cdot \frac{x}{s}, 0.5 \cdot \left|x\right|\right)}{s}, 2, \frac{\left|x\right|}{s}\right)}}{s} \]
13. Add Preprocessing

Alternative 9: 96.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 95.9

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

5. Applied rewrites 95.9%

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

6. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 96.2

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

8. Applied rewrites 96.2%

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

9. Final simplification 96.2%

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

Alternative 10: 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 99.8%

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

3. Taylor expanded in s around inf

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

   1. lower-*.f32 95.0

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

5. Applied rewrites 95.0%

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

Alternative 11: 81.2% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Add Preprocessing
3. 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. lift-*.f32 N/A

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

   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. associate-*l* N/A

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

   6. associate-/l* N/A

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

   7. associate-/r* N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-+l+ N/A

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

   4. distribute-lft-out N/A

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

   5. lower-fma.f32 N/A

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

7. Applied rewrites 22.6%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 81.8%

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

11. Final simplification 81.8%

    \[\leadsto \frac{\frac{1}{s}}{\frac{x}{s \cdot s} \cdot x + 4} \]
12. Add Preprocessing

Alternative 12: 26.9% accurate, 31.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 27.0

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

5. Applied rewrites 27.0%

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

Reproduce

herbie shell --seed 2024235 
(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))))))