Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 13.2s

Alternatives: 8

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 96.7% accurate, 0.9× speedup

Math

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

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)
     0.0
     (/ (fma (/ (* x (/ x s)) s) -0.0625 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 = 0.0f;
	} else {
		tmp = fmaf(((x * (x / s)) / s), -0.0625f, 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(0.0);
	else
		tmp = Float32(fma(Float32(Float32(x * Float32(x / s)) / s), Float32(-0.0625), 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 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

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{{s}^{2}} + \frac{1}{24} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right) - \left(\frac{1}{4} + \left(\frac{1}{16} \cdot \frac{\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + {\left(\left|x\right|\right)}^{3}}{{s}^{3}} + \frac{\left|x\right| \cdot \left(\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{{s}^{3}}\right)\right)}{s}} \]

   5. Simplified 0.2%

      \[\leadsto \color{blue}{\frac{\frac{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}{s \cdot \left(s \cdot s\right)} - \left(\frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot s} + \mathsf{fma}\left(0.10416666666666667, \frac{\left|x\right| \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, \mathsf{fma}\left(\left|x\right|, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(s \cdot s\right)}, 0.25\right)\right)\right)}{-s}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right) - \left(\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)\right)}{{s}^{4}}} \]
   7. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}\right)} \]

      2. *-commutative N/A

         \[\leadsto -1 \cdot \left(\frac{\color{blue}{\left({x}^{2} \cdot \left|x\right|\right) \cdot \frac{1}{24}}}{{s}^{4}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}\right) \]

      3. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\frac{\left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\left(\frac{-1}{16} + \frac{5}{48}\right)}}{{s}^{4}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto -1 \cdot \left(\frac{\color{blue}{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}}{{s}^{4}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}\right) \]

      5. +-inverses N/A

         \[\leadsto -1 \cdot \color{blue}{0} \]

      6. metadata-eval 99.9

         \[\leadsto \color{blue}{0} \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{0} \]

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{{s}^{2}} + \frac{1}{24} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right) - \left(\frac{1}{4} + \left(\frac{1}{16} \cdot \frac{\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + {\left(\left|x\right|\right)}^{3}}{{s}^{3}} + \frac{\left|x\right| \cdot \left(\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{{s}^{3}}\right)\right)}{s}} \]

   5. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{\frac{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}{s \cdot \left(s \cdot s\right)} - \left(\frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot s} + \mathsf{fma}\left(0.10416666666666667, \frac{\left|x\right| \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, \mathsf{fma}\left(\left|x\right|, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(s \cdot s\right)}, 0.25\right)\right)\right)}{-s}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)}{s \cdot \left(s \cdot s\right)} - \color{blue}{\left(\frac{1}{4} + {x}^{2} \cdot \left(\left(\frac{-1}{16} \cdot \frac{\left|x\right|}{{s}^{3}} + \frac{5}{48} \cdot \frac{\left|x\right|}{{s}^{3}}\right) - \frac{1}{16} \cdot \frac{1}{{s}^{2}}\right)\right)}}{\mathsf{neg}\left(s\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\frac{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)}{s \cdot \left(s \cdot s\right)} - \color{blue}{\left({x}^{2} \cdot \left(\left(\frac{-1}{16} \cdot \frac{\left|x\right|}{{s}^{3}} + \frac{5}{48} \cdot \frac{\left|x\right|}{{s}^{3}}\right) - \frac{1}{16} \cdot \frac{1}{{s}^{2}}\right) + \frac{1}{4}\right)}}{\mathsf{neg}\left(s\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{\frac{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)}{s \cdot \left(s \cdot s\right)} - \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(\frac{-1}{16} \cdot \frac{\left|x\right|}{{s}^{3}} + \frac{5}{48} \cdot \frac{\left|x\right|}{{s}^{3}}\right) - \frac{1}{16} \cdot \frac{1}{{s}^{2}}\right) + \frac{1}{4}\right)}{\mathsf{neg}\left(s\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\frac{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)}{s \cdot \left(s \cdot s\right)} - \left(\color{blue}{x \cdot \left(x \cdot \left(\left(\frac{-1}{16} \cdot \frac{\left|x\right|}{{s}^{3}} + \frac{5}{48} \cdot \frac{\left|x\right|}{{s}^{3}}\right) - \frac{1}{16} \cdot \frac{1}{{s}^{2}}\right)\right)} + \frac{1}{4}\right)}{\mathsf{neg}\left(s\right)} \]

      4. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{\frac{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{24}\right)}{s \cdot \left(s \cdot s\right)} - \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\left(\frac{-1}{16} \cdot \frac{\left|x\right|}{{s}^{3}} + \frac{5}{48} \cdot \frac{\left|x\right|}{{s}^{3}}\right) - \frac{1}{16} \cdot \frac{1}{{s}^{2}}\right), \frac{1}{4}\right)}}{\mathsf{neg}\left(s\right)} \]

   8. Simplified 61.4%

      \[\leadsto \frac{\frac{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}{s \cdot \left(s \cdot s\right)} - \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(0.041666666666666664, \frac{\left|x\right|}{s \cdot \left(s \cdot s\right)}, \frac{-0.0625}{s \cdot s}\right), 0.25\right)}}{-s} \]

   9. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{4} + \frac{-1}{16} \cdot \frac{{x}^{2}}{{s}^{2}}}{s}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{4} + \frac{-1}{16} \cdot \frac{{x}^{2}}{{s}^{2}}}{s}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{16} \cdot \frac{{x}^{2}}{{s}^{2}} + \frac{1}{4}}}{s} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{{s}^{2}} \cdot \frac{-1}{16}} + \frac{1}{4}}{s} \]

       4. accelerator-lowering-fma.f32 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{{x}^{2}}{{s}^{2}}, \frac{-1}{16}, \frac{1}{4}\right)}}{s} \]

       5. /-lowering-/.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{{x}^{2}}{{s}^{2}}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

       6. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{{s}^{2}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{{s}^{2}}, \frac{-1}{16}, \frac{1}{4}\right)}{s} \]

       8. unpow2 N/A

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

       9. *-lowering-*.f32 79.4

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

   11. Simplified 79.4%

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

       1. times-frac N/A

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

       2. associate-*r/ N/A

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

       3. /-lowering-/.f32 N/A

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

       4. *-lowering-*.f32 N/A

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

       5. /-lowering-/.f32 95.9

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

   13. Applied egg-rr 95.9%

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

4. Final simplification 98.8%

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

Alternative 3: 96.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- (/ (fabs x) s)))) (t_1 (+ t_0 1.0)))
   (if (<= (/ t_0 (* t_1 (* s t_1))) 0.0) 0.0 (/ 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 = 0.0f;
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = exp(-(abs(x) / s))
    t_1 = t_0 + 1.0e0
    if ((t_0 / (t_1 * (s * t_1))) <= 0.0e0) then
        tmp = 0.0e0
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(-Float32(abs(x) / s)))
	t_1 = Float32(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(0.0);
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 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

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{{s}^{2}} + \frac{1}{24} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right) - \left(\frac{1}{4} + \left(\frac{1}{16} \cdot \frac{\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + {\left(\left|x\right|\right)}^{3}}{{s}^{3}} + \frac{\left|x\right| \cdot \left(\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{{s}^{3}}\right)\right)}{s}} \]

   5. Simplified 0.2%

      \[\leadsto \color{blue}{\frac{\frac{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}{s \cdot \left(s \cdot s\right)} - \left(\frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot s} + \mathsf{fma}\left(0.10416666666666667, \frac{\left|x\right| \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, \mathsf{fma}\left(\left|x\right|, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(s \cdot s\right)}, 0.25\right)\right)\right)}{-s}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right) - \left(\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)\right)}{{s}^{4}}} \]
   7. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}\right)} \]

      2. *-commutative N/A

         \[\leadsto -1 \cdot \left(\frac{\color{blue}{\left({x}^{2} \cdot \left|x\right|\right) \cdot \frac{1}{24}}}{{s}^{4}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}\right) \]

      3. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\frac{\left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\left(\frac{-1}{16} + \frac{5}{48}\right)}}{{s}^{4}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}\right) \]

      4. distribute-rgt-out N/A

         \[\leadsto -1 \cdot \left(\frac{\color{blue}{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}}{{s}^{4}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}\right) \]

      5. +-inverses N/A

         \[\leadsto -1 \cdot \color{blue}{0} \]

      6. metadata-eval 99.9

         \[\leadsto \color{blue}{0} \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{0} \]

   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

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

      1. /-lowering-/.f32 93.5

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

   5. Simplified 93.5%

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

4. Final simplification 98.1%

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

Alternative 4: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. *-lowering-*.f32 N/A

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

   \[\leadsto e^{-\frac{\left|x\right|}{s}} \cdot \frac{{\left(e^{-\frac{\left|x\right|}{s}} + 1\right)}^{-2}}{s} \]
6. 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(-Float32(abs(x) / s))
	return Float32(exp(fma(Float32(-2.0), log1p(exp(t_0)), t_0)) / s)
end

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.9%

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

   1. /-lowering-/.f32 N/A

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

6. Applied egg-rr 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: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\mathsf{fma}\left(-\frac{x \cdot 0.25}{s}, \frac{x}{s}, -2 \cdot \log 2\right)}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (exp (fma (- (/ (* x 0.25) s)) (/ x s) (* -2.0 (log 2.0)))) s))

C

float code(float x, float s) {
	return expf(fmaf(-((x * 0.25f) / s), (x / s), (-2.0f * logf(2.0f)))) / s;
}

Julia

function code(x, s)
	return Float32(exp(fma(Float32(-Float32(Float32(x * Float32(0.25)) / s)), Float32(x / s), Float32(Float32(-2.0) * log(Float32(2.0))))) / s)
end

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.9%

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

   1. /-lowering-/.f32 N/A

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

6. Applied egg-rr 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. Taylor expanded in s around inf

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

   1. accelerator-lowering-fma.f32 N/A

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

   2. log-lowering-log.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-neg-frac2 N/A

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

   5. /-lowering-/.f32 N/A

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

   6. distribute-rgt-out N/A

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

   7. unpow2 N/A

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

   8. sqr-abs N/A

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

   9. unpow2 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

   17. metadata-eval N/A

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

   18. *-lowering-*.f32 N/A

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

   19. unpow2 N/A

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

   20. *-lowering-*.f32 N/A

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

   21. neg-lowering-neg.f32 N/A

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

   22. unpow2 N/A

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

   23. *-lowering-*.f32 90.3

       \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \log 2, \frac{0.25 \cdot \left(x \cdot x\right)}{-\color{blue}{s \cdot s}}\right)}}{s} \]

9. Simplified 90.3%

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. distribute-lft-neg-in N/A

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

    4. times-frac N/A

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

    5. accelerator-lowering-fma.f32 N/A

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

    6. /-lowering-/.f32 N/A

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

    7. *-commutative N/A

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

    8. *-lowering-*.f32 N/A

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

    9. neg-lowering-neg.f32 N/A

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

    10. /-lowering-/.f32 N/A

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

    11. *-lowering-*.f32 N/A

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

    12. log-lowering-log.f32 98.9

        \[\leadsto \frac{e^{\mathsf{fma}\left(\frac{x \cdot 0.25}{-s}, \frac{x}{s}, -2 \cdot \color{blue}{\log 2}\right)}}{s} \]

11. Applied egg-rr 98.9%

    \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(\frac{x \cdot 0.25}{-s}, \frac{x}{s}, -2 \cdot \log 2\right)}}}{s} \]

12. Final simplification 98.9%

    \[\leadsto \frac{e^{\mathsf{fma}\left(-\frac{x \cdot 0.25}{s}, \frac{x}{s}, -2 \cdot \log 2\right)}}{s} \]
13. Add Preprocessing

Alternative 7: 94.5% 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 99.7%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 96.2

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

5. Simplified 96.2%

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

6. Final simplification 96.2%

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

Alternative 8: 73.0% accurate, 373.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 0.0)

C

float code(float x, float s) {
	return 0.0f;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

function code(x, s)
	return Float32(0.0)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \frac{\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)}{{s}^{2}} + \frac{1}{24} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{{s}^{3}}\right) - \left(\frac{1}{4} + \left(\frac{1}{16} \cdot \frac{\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + {\left(\left|x\right|\right)}^{3}}{{s}^{3}} + \frac{\left|x\right| \cdot \left(\frac{1}{8} \cdot {\left(\left|x\right|\right)}^{2} - \frac{1}{16} \cdot \left(2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}\right)\right)}{{s}^{3}}\right)\right)}{s}} \]

5. Simplified 17.4%

   \[\leadsto \color{blue}{\frac{\frac{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot 0.041666666666666664\right)}{s \cdot \left(s \cdot s\right)} - \left(\frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot s} + \mathsf{fma}\left(0.10416666666666667, \frac{\left|x\right| \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, \mathsf{fma}\left(\left|x\right|, \frac{\left(x \cdot x\right) \cdot -0.0625}{s \cdot \left(s \cdot s\right)}, 0.25\right)\right)\right)}{-s}} \]

6. Taylor expanded in s around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right) - \left(\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)\right)}{{s}^{4}}} \]
7. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{\frac{1}{24} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}\right)} \]

   2. *-commutative N/A

      \[\leadsto -1 \cdot \left(\frac{\color{blue}{\left({x}^{2} \cdot \left|x\right|\right) \cdot \frac{1}{24}}}{{s}^{4}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}\right) \]

   3. metadata-eval N/A

      \[\leadsto -1 \cdot \left(\frac{\left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\left(\frac{-1}{16} + \frac{5}{48}\right)}}{{s}^{4}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}\right) \]

   4. distribute-rgt-out N/A

      \[\leadsto -1 \cdot \left(\frac{\color{blue}{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}}{{s}^{4}} - \frac{\frac{-1}{16} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{5}{48} \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{s}^{4}}\right) \]

   5. +-inverses N/A

      \[\leadsto -1 \cdot \color{blue}{0} \]

   6. metadata-eval 73.3

      \[\leadsto \color{blue}{0} \]

8. Simplified 73.3%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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))))))