Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 13.0s

Alternatives: 10

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	return Float32(t_0 / Float32(fma(t_0, s, s) * Float32(t_0 + Float32(1.0))))
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-fabs.f32 N/A

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

   2. remove-double-neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. frac-2neg N/A

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

   6. frac-2neg N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f32 99.8

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

   13. lift-/.f32 N/A

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

   14. lift-neg.f32 N/A

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

   15. distribute-frac-neg N/A

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

   16. lower-neg.f32 N/A

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

   17. lower-/.f32 99.8

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

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%

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

Alternative 2: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 0:\\ \;\;\;\;\frac{t\_0}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x) (- s)))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 0.0)
     (/ t_0 s)
     (/ (fma (/ x s) (/ (* x -0.0625) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 0.0f) {
		tmp = t_0 / s;
	} else {
		tmp = fmaf((x / s), ((x * -0.0625f) / s), 0.25f) / s;
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-fabs.f32 N/A

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

      2. remove-double-neg N/A

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

      3. lift-neg.f32 N/A

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

      4. remove-double-neg N/A

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

      5. frac-2neg N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f32 N/A

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

      8. lift-exp.f32 N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-in N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f32 100.0

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

      13. lift-/.f32 N/A

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

      14. lift-neg.f32 N/A

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

      15. distribute-frac-neg N/A

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

      16. lower-neg.f32 N/A

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

      17. lower-/.f32 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in s around 0

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

      1. neg-mul-1 N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower-fabs.f32 100.0

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

   9. Applied rewrites 100.0%

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

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

   1. Initial program 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

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{4} + \frac{1}{8} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) - \frac{1}{16} \cdot \frac{2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{{s}^{2}}}{s}} \]
   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 70.9%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f32 N/A

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

      11. times-frac N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-/.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 91.9

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

   7. Applied rewrites 91.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\left|x\right|}{-s}}}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \left(s \cdot \left(e^{\frac{\left|x\right|}{-s}} + 1\right)\right)} \leq 0:\\ \;\;\;\;\frac{e^{\frac{\left|x\right|}{-s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ t_1 := t\_0 + 1\\ \mathbf{if}\;\frac{t\_0}{t\_1 \cdot \left(s \cdot t\_1\right)} \leq 200000000753664:\\ \;\;\;\;\frac{\frac{-0.0625}{x \cdot \left(x \cdot \frac{-0.0625}{s \cdot s}\right) - 0.25}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x) (- s)))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 200000000753664.0)
     (/ (/ -0.0625 (- (* x (* x (/ -0.0625 (* s s)))) 0.25)) s)
     (/ (fma (/ x s) (/ (* x -0.0625) s) 0.25) s))))

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	float t_1 = t_0 + 1.0f;
	float tmp;
	if ((t_0 / (t_1 * (s * t_1))) <= 200000000753664.0f) {
		tmp = (-0.0625f / ((x * (x * (-0.0625f / (s * s)))) - 0.25f)) / s;
	} else {
		tmp = fmaf((x / s), ((x * -0.0625f) / s), 0.25f) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(200000000753664.0))
		tmp = Float32(Float32(Float32(-0.0625) / Float32(Float32(x * Float32(x * Float32(Float32(-0.0625) / Float32(s * s)))) - Float32(0.25))) / s);
	else
		tmp = Float32(fma(Float32(x / s), Float32(Float32(x * Float32(-0.0625)) / s), Float32(0.25)) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) < 2.00000001e14

   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. 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 20.4%

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

   7. Applied rewrites 14.1%

      \[\leadsto \frac{\color{blue}{\frac{\frac{0.00390625 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{\left(s \cdot s\right) \cdot \left(s \cdot s\right)} - 0.0625}{x \cdot \left(x \cdot \frac{-0.0625}{s \cdot s}\right) - 0.25}}}{s} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 88.4%

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

   if 2.00000001e14 < (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))

   1. Initial program 98.6%

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

   3. Taylor expanded in s around inf

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

      1. 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 29.5%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f32 N/A

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

      11. times-frac N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-/.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower-*.f32 83.2

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

   7. Applied rewrites 83.2%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\frac{\left|x\right|}{-s}}}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \left(s \cdot \left(e^{\frac{\left|x\right|}{-s}} + 1\right)\right)} \leq 200000000753664:\\ \;\;\;\;\frac{\frac{-0.0625}{x \cdot \left(x \cdot \frac{-0.0625}{s \cdot s}\right) - 0.25}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{s}, \frac{x \cdot -0.0625}{s}, 0.25\right)}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(exp(fma(Float32(Float32(-1.0) / s), abs(x), Float32(Float32(-2.0) * log1p(exp(Float32(abs(x) / Float32(-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-fabs.f32 N/A

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

   2. remove-double-neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. frac-2neg N/A

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

   6. frac-2neg N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f32 99.8

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

   13. lift-/.f32 N/A

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

   14. lift-neg.f32 N/A

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

   15. distribute-frac-neg N/A

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

   16. lower-neg.f32 N/A

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

   17. lower-/.f32 99.8

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

   1. lift-fabs.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-log1p.f32 N/A

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

   6. lift-fabs.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-/.f32 N/A

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

   11. clear-num N/A

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

   12. associate-/r/ N/A

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

   13. lower-fma.f32 N/A

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

8. Applied rewrites 99.8%

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

Alternative 5: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. lift-fabs.f32 N/A

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

   2. remove-double-neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. frac-2neg N/A

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

   6. frac-2neg N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f32 99.8

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

   13. lift-/.f32 N/A

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

   14. lift-neg.f32 N/A

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

   15. distribute-frac-neg N/A

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

   16. lower-neg.f32 N/A

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

   17. lower-/.f32 99.8

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

Alternative 6: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 96.8

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

5. Applied rewrites 96.8%

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

6. Final simplification 96.8%

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

Alternative 7: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 96.8

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

5. Applied rewrites 96.8%

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

6. Taylor expanded in s around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-fabs.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. mul-1-neg N/A

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

   16. lower-neg.f32 N/A

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

   17. lower-fabs.f32 96.8

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

8. Applied rewrites 96.8%

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

9. Final simplification 96.8%

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

Alternative 8: 95.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	t_0 = exp(Float32(abs(x) / Float32(-s)))
	return Float32(t_0 / Float32(fma(t_0, s, s) * Float32(2.0)))
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-fabs.f32 N/A

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

   2. remove-double-neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. frac-2neg N/A

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

   6. frac-2neg N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-in N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f32 99.8

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

   13. lift-/.f32 N/A

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

   14. lift-neg.f32 N/A

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

   15. distribute-frac-neg N/A

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

   16. lower-neg.f32 N/A

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

   17. lower-/.f32 99.8

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

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. Taylor expanded in s around inf

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

7. Applied rewrites 95.4%

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

8. Final simplification 95.4%

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

Alternative 9: 94.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 95.1

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

5. Applied rewrites 95.1%

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

6. Final simplification 95.1%

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

Alternative 10: 26.4% 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 27.3

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

5. Applied rewrites 27.3%

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

Reproduce

herbie shell --seed 2024219 
(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))))))