Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 12.9s

Alternatives: 9

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(-Float32(abs(x) / 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-*.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

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. lower-neg.f32 N/A

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

   11. 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(-Float32(abs(x) / s)))
	t_1 = Float32(t_0 + Float32(1.0))
	tmp = Float32(0.0)
	if (Float32(t_0 / Float32(t_1 * Float32(s * t_1))) <= Float32(0.0))
		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-*.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 100.0

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

      7. lift-/.f32 N/A

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

      8. lift-neg.f32 N/A

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

      9. distribute-frac-neg N/A

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

      10. lower-neg.f32 N/A

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

      11. 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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in s around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. lower-+.f32 N/A

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

      6. lower-/.f32 N/A

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

   7. Applied rewrites 70.9%

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

   9. Applied rewrites 91.9%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{x \cdot -0.0625}{s}}, 0.25\right)}{s} \]
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: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(exp(fma(abs(x), Float32(Float32(-1.0) / s), Float32(Float32(-2.0) * log1p(exp(Float32(-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

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. lower-neg.f32 N/A

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

   11. 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-fma.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. distribute-frac-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. clear-num N/A

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

   8. associate-/r/ N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 99.8

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

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. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. div-inv N/A

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

   5. lift-neg.f32 N/A

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

   6. distribute-frac-neg N/A

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

   7. distribute-neg-frac2 N/A

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

   8. div-inv N/A

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

   9. lower-fma.f32 N/A

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

   10. metadata-eval N/A

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

   11. frac-2neg N/A

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

   12. lower-/.f32 99.8

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

10. Applied rewrites 99.8%

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

Alternative 4: 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(-Float32(abs(x) / 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-*.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

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. lower-neg.f32 N/A

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

   11. 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 5: 95.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ t_1 := e^{-t\_0}\\ \frac{t\_1}{\left(s \cdot \left(t\_1 + 1\right)\right) \cdot \left(2 - t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) s)) (t_1 (exp (- t_0))))
   (/ t_1 (* (* s (+ t_1 1.0)) (- 2.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / s;
	float t_1 = expf(-t_0);
	return t_1 / ((s * (t_1 + 1.0f)) * (2.0f - t_0));
}

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 = abs(x) / s
    t_1 = exp(-t_0)
    code = t_1 / ((s * (t_1 + 1.0e0)) * (2.0e0 - t_0))
end function

Julia

function code(x, s)
	t_0 = Float32(abs(x) / s)
	t_1 = exp(Float32(-t_0))
	return Float32(t_1 / Float32(Float32(s * Float32(t_1 + Float32(1.0))) * Float32(Float32(2.0) - t_0)))
end

MATLAB

function tmp = code(x, s)
	t_0 = abs(x) / s;
	t_1 = exp(-t_0);
	tmp = t_1 / ((s * (t_1 + single(1.0))) * (single(2.0) - t_0));
end

Derivation

1. Initial program 99.8%

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

3. 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 6: 95.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ t_1 := e^{-t\_0}\\ \frac{t\_1}{\mathsf{fma}\left(t\_1, s, s\right) \cdot \left(2 - t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) s)) (t_1 (exp (- t_0))))
   (/ t_1 (* (fma t_1 s s) (- 2.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / s;
	float t_1 = expf(-t_0);
	return t_1 / (fmaf(t_1, s, s) * (2.0f - t_0));
}

Julia

function code(x, s)
	t_0 = Float32(abs(x) / s)
	t_1 = exp(Float32(-t_0))
	return Float32(t_1 / Float32(fma(t_1, s, s) * Float32(Float32(2.0) - t_0)))
end

Derivation

1. Initial program 99.8%

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

   1. 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

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. lower-neg.f32 N/A

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

   11. 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

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-fabs.f32 96.8

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

7. Applied rewrites 96.8%

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

8. Final simplification 96.8%

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

Alternative 7: 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(-Float32(abs(x) / 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-*.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

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. lower-neg.f32 N/A

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

   11. 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 8: 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(-Float32(abs(x) / s))) / Float32(s * Float32(4.0)))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. 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 9: 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))))))