Logistic distribution

Percentage Accurate: 99.5% → 99.4%

Time: 12.8s

Alternatives: 11

Speedup: 2.9×

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: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(\cosh \left(\frac{\left|x\right|}{s}\right)\right)} \cdot \frac{0.5}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (* (exp (- (log1p (cosh (/ (fabs x) s))))) (/ 0.5 s)))

C

float code(float x, float s) {
	return expf(-log1pf(coshf((fabsf(x) / s)))) * (0.5f / s);
}

Julia

function code(x, s)
	return Float32(exp(Float32(-log1p(cosh(Float32(abs(x) / s))))) * Float32(Float32(0.5) / s))
end

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(s \cdot \left(\left(e^{\frac{0 - \left|x\right|}{s}} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(e^{\frac{0 - \left|x\right|}{s}} + e^{\frac{\left|x\right|}{s}}\right) \cdot s + \color{blue}{2 \cdot s}\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\left(e^{\frac{0 - \left|x\right|}{s}} + e^{\frac{\left|x\right|}{s}}\right) \cdot s\right), \color{blue}{\left(2 \cdot s\right)}\right)\right) \]

6. Applied egg-rr 99.4%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-/r* N/A

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

   4. div-inv N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt1-in N/A

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

   7. times-frac N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{1}{\cosh \left(\frac{\left|x\right|}{s}\right) + 1}\right), \color{blue}{\left(\frac{\frac{1}{s}}{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \left(\cosh \left(\frac{\left|x\right|}{s}\right) + 1\right)\right), \left(\frac{\color{blue}{\frac{1}{s}}}{2}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\cosh \left(\frac{\left|x\right|}{s}\right), 1\right)\right), \left(\frac{\frac{1}{\color{blue}{s}}}{2}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right), 1\right)\right), \left(\frac{\frac{1}{s}}{2}\right)\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right), 1\right)\right), \left(\frac{\frac{1}{s}}{2}\right)\right) \]

   13. fabs-lowering-fabs.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right), 1\right)\right), \left(\frac{\frac{1}{s}}{2}\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right), 1\right)\right), \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{2}\right)\right) \]

   15. /-lowering-/.f32 99.4%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right), 1\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), 2\right)\right) \]

8. Applied egg-rr 99.4%

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

   1. inv-pow N/A

      \[\leadsto \mathsf{*.f32}\left(\left({\left(\cosh \left(\frac{\left|x\right|}{s}\right) + 1\right)}^{-1}\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, s\right)}, 2\right)\right) \]

   2. pow-to-exp N/A

      \[\leadsto \mathsf{*.f32}\left(\left(e^{\log \left(\cosh \left(\frac{\left|x\right|}{s}\right) + 1\right) \cdot -1}\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, s\right)}, 2\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\left(\log \left(\cosh \left(\frac{\left|x\right|}{s}\right) + 1\right) \cdot -1\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{/.f32}\left(1, s\right)}, 2\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\log \left(\cosh \left(\frac{\left|x\right|}{s}\right) + 1\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{1}, s\right), 2\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\log \left(1 + \cosh \left(\frac{\left|x\right|}{s}\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), 2\right)\right) \]

   6. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{log1p}\left(\cosh \left(\frac{\left|x\right|}{s}\right)\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), 2\right)\right) \]

   7. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\cosh \left(\frac{\left|x\right|}{s}\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), 2\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), 2\right)\right) \]

   10. fabs-lowering-fabs.f32 99.4%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right), -1\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), 2\right)\right) \]

10. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh \left(\frac{\left|x\right|}{s}\right)\right) \cdot -1}} \cdot \frac{\frac{1}{s}}{2} \]
11. Step-by-step derivation

    1. associate-/l/ N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right), -1\right)\right), \left(\frac{1}{\color{blue}{2 \cdot s}}\right)\right) \]

    2. associate-/r* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right), -1\right)\right), \left(\frac{\frac{1}{2}}{\color{blue}{s}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right), -1\right)\right), \mathsf{/.f32}\left(\left(\frac{1}{2}\right), \color{blue}{s}\right)\right) \]

    4. metadata-eval 99.4%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right), -1\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, s\right)\right) \]

12. Applied egg-rr 99.4%

    \[\leadsto e^{\mathsf{log1p}\left(\cosh \left(\frac{\left|x\right|}{s}\right)\right) \cdot -1} \cdot \color{blue}{\frac{0.5}{s}} \]

13. Final simplification 99.4%

    \[\leadsto e^{-\mathsf{log1p}\left(\cosh \left(\frac{\left|x\right|}{s}\right)\right)} \cdot \frac{0.5}{s} \]
14. Add Preprocessing

Alternative 2: 99.6% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ (* s (* (cosh (/ (fabs x) s)) 2.0)) (* s 2.0))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(s \cdot \left(\left(e^{\frac{0 - \left|x\right|}{s}} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(e^{\frac{0 - \left|x\right|}{s}} + e^{\frac{\left|x\right|}{s}}\right) \cdot s + \color{blue}{2 \cdot s}\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\left(e^{\frac{0 - \left|x\right|}{s}} + e^{\frac{\left|x\right|}{s}}\right) \cdot s\right), \color{blue}{\left(2 \cdot s\right)}\right)\right) \]

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

   \[\leadsto \frac{1}{s \cdot \left(\cosh \left(\frac{\left|x\right|}{s}\right) \cdot 2\right) + s \cdot 2} \]
8. Add Preprocessing

Alternative 3: 99.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{s}}{\cosh \left(\frac{\left|x\right|}{s}\right) + 1} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (/ 0.5 s) (+ (cosh (/ (fabs x) s)) 1.0)))

C

float code(float x, float s) {
	return (0.5f / s) / (coshf((fabsf(x) / s)) + 1.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (0.5e0 / s) / (cosh((abs(x) / s)) + 1.0e0)
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(0.5) / s) / Float32(cosh(Float32(abs(x) / s)) + Float32(1.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(0.5) / s) / (cosh((abs(x) / s)) + single(1.0));
end

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}} \]

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

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{s}\right), \color{blue}{\left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\color{blue}{e^{\frac{0 - \left|x\right|}{s}}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)\right) \]

   4. associate-+r+ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(\left(e^{\frac{0 - \left|x\right|}{s}} + e^{\frac{\left|x\right|}{s}}\right) + \color{blue}{2}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \left(2 + \color{blue}{\left(e^{\frac{0 - \left|x\right|}{s}} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \color{blue}{\left(e^{\frac{0 - \left|x\right|}{s}} + e^{\frac{\left|x\right|}{s}}\right)}\right)\right) \]

   7. div-inv N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\left(0 - \left|x\right|\right) \cdot \frac{1}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\left(\mathsf{neg}\left(\left|x\right|\right)\right) \cdot \frac{1}{s}} + e^{\frac{\left|\color{blue}{x}\right|}{s}}\right)\right)\right) \]

   9. div-inv N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}} + e^{\frac{\color{blue}{\left|x\right|}}{s}}\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + \color{blue}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right)\right)\right) \]

   11. distribute-frac-neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(e^{\frac{\left|x\right|}{s}} + e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

   12. cosh-undef N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \left(2 \cdot \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right)\right) \]

   14. frac-2neg N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \cosh \left(\frac{\mathsf{neg}\left(\left|x\right|\right)}{\mathsf{neg}\left(s\right)}\right)\right)\right)\right) \]

   15. distribute-frac-neg2 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \cosh \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}\right)\right)\right)\right)\right) \]

   16. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, s\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(2, \cosh \left(0 - \frac{\mathsf{neg}\left(\left|x\right|\right)}{s}\right)\right)\right)\right) \]

6. Applied egg-rr 99.4%

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

   1. *-commutative N/A

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

   2. distribute-rgt1-in N/A

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

   3. div-inv N/A

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

   4. frac-times N/A

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

   5. *-commutative N/A

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

   6. un-div-inv N/A

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

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

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{1}{s}}{2}\right), \color{blue}{\left(\cosh \left(\frac{\left|x\right|}{s}\right) + 1\right)}\right) \]

   8. associate-/l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{2 \cdot s}\right), \left(\color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)} + 1\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{1}{2}}{s}\right), \left(\color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)} + 1\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{1}{2}}{s}\right), \left(\cosh \color{blue}{\left(\frac{\left|x\right|}{s}\right)} + 1\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \left(\color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)} + 1\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \left(1 + \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right) \]

   13. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \mathsf{+.f32}\left(1, \color{blue}{\cosh \left(\frac{\left|x\right|}{s}\right)}\right)\right) \]

   14. cosh-lowering-cosh.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \mathsf{+.f32}\left(1, \mathsf{cosh.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right)\right) \]

   15. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \mathsf{+.f32}\left(1, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right)\right) \]

   16. fabs-lowering-fabs.f32 99.4%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \mathsf{+.f32}\left(1, \mathsf{cosh.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right)\right) \]

8. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{\frac{0.5}{s}}{1 + \cosh \left(\frac{\left|x\right|}{s}\right)}} \]

9. Final simplification 99.4%

   \[\leadsto \frac{\frac{0.5}{s}}{\cosh \left(\frac{\left|x\right|}{s}\right) + 1} \]
10. Add Preprocessing

Alternative 4: 94.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in s around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \color{blue}{\left(4 \cdot s\right)}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \left(s \cdot \color{blue}{4}\right)\right) \]

   2. *-lowering-*.f32 95.5%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{*.f32}\left(s, \color{blue}{4}\right)\right) \]

5. Simplified 95.5%

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

6. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{-1 \cdot \frac{\left|x\right|}{s}}}{\color{blue}{s}} \]

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

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{4} \cdot e^{-1 \cdot \frac{\left|x\right|}{s}}\right), \color{blue}{s}\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(\frac{\left|x\right|}{s}\right)}\right), s\right) \]

   4. rec-exp N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{1}{4} \cdot \frac{1}{e^{\frac{\left|x\right|}{s}}}\right), s\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{1}{4} \cdot 1}{e^{\frac{\left|x\right|}{s}}}\right), s\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\left(\frac{\frac{1}{4}}{e^{\frac{\left|x\right|}{s}}}\right), s\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{4}, \left(e^{\frac{\left|x\right|}{s}}\right)\right), s\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{4}, \mathsf{exp.f32}\left(\left(\frac{\left|x\right|}{s}\right)\right)\right), s\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{4}, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), s\right)\right)\right), s\right) \]

   10. fabs-lowering-fabs.f32 95.5%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\frac{1}{4}, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), s\right)\right)\right), s\right) \]

8. Simplified 95.5%

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

Alternative 5: 83.6% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot \frac{x}{s \cdot s} + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(s \cdot \left(s \cdot -4\right)\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999987376214e-7)
   (/ 1.0 (* s (+ (* x (/ x (* s s))) 4.0)))
   (/ (* s (* s (* s -4.0))) (* (* x x) (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999987376214e-7f) {
		tmp = 1.0f / (s * ((x * (x / (s * s))) + 4.0f));
	} else {
		tmp = (s * (s * (s * -4.0f))) / ((x * x) * (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.999999987376214e-7) then
        tmp = 1.0e0 / (s * ((x * (x / (s * s))) + 4.0e0))
    else
        tmp = (s * (s * (s * (-4.0e0)))) / ((x * x) * (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.999999987376214e-7))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(x * Float32(x / Float32(s * s))) + Float32(4.0))));
	else
		tmp = Float32(Float32(s * Float32(s * Float32(s * Float32(-4.0)))) / Float32(Float32(x * x) * Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.999999987376214e-7))
		tmp = single(1.0) / (s * ((x * (x / (s * s))) + single(4.0)));
	else
		tmp = (s * (s * (s * single(-4.0)))) / ((x * x) * (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999e-7

   1. Initial program 99.1%

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

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(s \cdot \left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right) + \color{blue}{4}\right)\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right) + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + 4\right)\right)\right) \]

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right)\right) \]

   7. Simplified 75.1%

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

      1. associate-/l/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{x \cdot x}{s \cdot s}\right), 4\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(x \cdot \frac{x}{s \cdot s}\right), 4\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{x}{s \cdot s}\right)\right), 4\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \left(s \cdot s\right)\right)\right), 4\right)\right)\right) \]

      5. *-lowering-*.f32 85.0%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \mathsf{*.f32}\left(s, s\right)\right)\right), 4\right)\right)\right) \]

   9. Applied egg-rr 85.0%

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

   if 4.99999999e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(s \cdot \left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right) + \color{blue}{4}\right)\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right) + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + 4\right)\right)\right) \]

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right)\right) \]

   7. Simplified 88.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{s + -4 \cdot \frac{{s}^{3}}{{x}^{2}}}{{x}^{2}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f32}\left(\left(s + -4 \cdot \frac{{s}^{3}}{{x}^{2}}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(\left(s + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{{s}^{3}}{{x}^{2}}\right), \left({x}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(\left(s + \left(\mathsf{neg}\left(4 \cdot \frac{{s}^{3}}{{x}^{2}}\right)\right)\right), \left({x}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \left(\mathsf{neg}\left(4 \cdot \frac{{s}^{3}}{{x}^{2}}\right)\right)\right), \left({\color{blue}{x}}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \left(\left(\mathsf{neg}\left(4\right)\right) \cdot \frac{{s}^{3}}{{x}^{2}}\right)\right), \left({x}^{2}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \left(-4 \cdot \frac{{s}^{3}}{{x}^{2}}\right)\right), \left({x}^{2}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \left(\frac{-4 \cdot {s}^{3}}{{x}^{2}}\right)\right), \left({x}^{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\left(-4 \cdot {s}^{3}\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

      9. unpow3 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\left(-4 \cdot \left(\left(s \cdot s\right) \cdot s\right)\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\left(-4 \cdot \left({s}^{2} \cdot s\right)\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\left(\left(-4 \cdot {s}^{2}\right) \cdot s\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(-4 \cdot {s}^{2}\right), s\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({s}^{2} \cdot -4\right), s\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\left({s}^{2}\right), -4\right), s\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\left(s \cdot s\right), -4\right), s\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right), s\right), \left({x}^{2}\right)\right)\right), \left({x}^{2}\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right), s\right), \left(x \cdot x\right)\right)\right), \left({x}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right), s\right), \mathsf{*.f32}\left(x, x\right)\right)\right), \left({x}^{2}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right), s\right), \mathsf{*.f32}\left(x, x\right)\right)\right), \left(x \cdot \color{blue}{x}\right)\right) \]

      20. *-lowering-*.f32 78.3%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{+.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -4\right), s\right), \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

   10. Simplified 78.3%

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

   11. Taylor expanded in s around inf

       \[\leadsto \color{blue}{-4 \cdot \frac{{s}^{3}}{{x}^{4}}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{-4 \cdot {s}^{3}}{\color{blue}{{x}^{4}}} \]

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

          \[\leadsto \mathsf{/.f32}\left(\left(-4 \cdot {s}^{3}\right), \color{blue}{\left({x}^{4}\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\left({s}^{3} \cdot -4\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       4. cube-mult N/A

          \[\leadsto \mathsf{/.f32}\left(\left(\left(s \cdot \left(s \cdot s\right)\right) \cdot -4\right), \left({x}^{4}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(\left(s \cdot {s}^{2}\right) \cdot -4\right), \left({x}^{4}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{/.f32}\left(\left(s \cdot \left({s}^{2} \cdot -4\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\left(s \cdot \left(-4 \cdot {s}^{2}\right)\right), \left({x}^{4}\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \left(-4 \cdot {s}^{2}\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \left({s}^{2} \cdot -4\right)\right), \left({x}^{4}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \left(\left(s \cdot s\right) \cdot -4\right)\right), \left({x}^{4}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \left(s \cdot \left(s \cdot -4\right)\right)\right), \left({x}^{4}\right)\right) \]

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

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \left(s \cdot -4\right)\right)\right), \left({x}^{4}\right)\right) \]

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

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, -4\right)\right)\right), \left({x}^{4}\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, -4\right)\right)\right), \left({x}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       15. pow-sqr N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, -4\right)\right)\right), \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right) \]

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

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, -4\right)\right)\right), \mathsf{*.f32}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, -4\right)\right)\right), \mathsf{*.f32}\left(\left(x \cdot x\right), \left({\color{blue}{x}}^{2}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, -4\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({\color{blue}{x}}^{2}\right)\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, -4\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       20. *-lowering-*.f32 95.0%

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, -4\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 95.0%

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

Alternative 6: 74.5% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 5.999999920033662 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s \cdot \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s} + s \cdot 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= s 5.999999920033662e-24)
   (/ 1.0 (* s (/ (* x x) (* s s))))
   (/ 1.0 (+ (/ (* x x) s) (* s 4.0)))))

C

float code(float x, float s) {
	float tmp;
	if (s <= 5.999999920033662e-24f) {
		tmp = 1.0f / (s * ((x * x) / (s * s)));
	} else {
		tmp = 1.0f / (((x * x) / s) + (s * 4.0f));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (s <= 5.999999920033662e-24) then
        tmp = 1.0e0 / (s * ((x * x) / (s * s)))
    else
        tmp = 1.0e0 / (((x * x) / s) + (s * 4.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (s <= Float32(5.999999920033662e-24))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(x * x) / Float32(s * s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(x * x) / s) + Float32(s * Float32(4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (s <= single(5.999999920033662e-24))
		tmp = single(1.0) / (s * ((x * x) / (s * s)));
	else
		tmp = single(1.0) / (((x * x) / s) + (s * single(4.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 5.99999992e-24

   1. Initial program 99.7%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(s \cdot \left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right) + \color{blue}{4}\right)\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right) + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + 4\right)\right)\right) \]

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right)\right) \]

   7. Simplified 69.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(\frac{{x}^{2}}{{s}^{2}}\right)}\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\left({x}^{2}\right), \color{blue}{\left({s}^{2}\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\left(x \cdot x\right), \left({\color{blue}{s}}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({\color{blue}{s}}^{2}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot \color{blue}{s}\right)\right)\right)\right) \]

      5. *-lowering-*.f32 82.7%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right) \]

   10. Simplified 82.7%

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

   if 5.99999992e-24 < s

   1. Initial program 99.1%

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

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(s \cdot \left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right) + \color{blue}{4}\right)\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right) + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + 4\right)\right)\right) \]

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right)\right) \]

   7. Simplified 84.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(4 \cdot s + \frac{{x}^{2}}{s}\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{{x}^{2}}{s} + \color{blue}{4 \cdot s}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{{x}^{2}}{s}\right), \color{blue}{\left(4 \cdot s\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left({x}^{2}\right), s\right), \left(\color{blue}{4} \cdot s\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot x\right), s\right), \left(4 \cdot s\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), s\right), \left(4 \cdot s\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), s\right), \left(s \cdot \color{blue}{4}\right)\right)\right) \]

      7. *-lowering-*.f32 80.5%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), s\right), \mathsf{*.f32}\left(s, \color{blue}{4}\right)\right)\right) \]

   10. Simplified 80.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s} + s \cdot 4}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 51.0% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.000000033724767 \cdot 10^{-16}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.000000033724767e-16)
   (/ 0.25 s)
   (/ 1.0 (* s (/ (* x x) (* s s))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.000000033724767e-16f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (s * ((x * x) / (s * s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 2.000000033724767e-16) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (s * ((x * x) / (s * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.000000033724767e-16))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(x * x) / Float32(s * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.000000033724767e-16))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (s * ((x * x) / (s * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.00000003e-16

   1. Initial program 99.1%

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

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 39.1%

         \[\leadsto \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right) \]

   7. Simplified 39.1%

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

   if 2.00000003e-16 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(s \cdot \left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right) + \color{blue}{4}\right)\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right) + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + 4\right)\right)\right) \]

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right)\right) \]

   7. Simplified 83.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(\frac{{x}^{2}}{{s}^{2}}\right)}\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\left({x}^{2}\right), \color{blue}{\left({s}^{2}\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\left(x \cdot x\right), \left({\color{blue}{s}}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left({\color{blue}{s}}^{2}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(s \cdot \color{blue}{s}\right)\right)\right)\right) \]

      5. *-lowering-*.f32 85.8%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right) \]

   10. Simplified 85.8%

       \[\leadsto \frac{1}{s \cdot \color{blue}{\frac{x \cdot x}{s \cdot s}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.6% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in s around inf

   \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(s \cdot \left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)\right)}\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right) + \color{blue}{4}\right)\right)\right) \]

   3. associate-+r+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right) + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + 4\right)\right)\right) \]

   4. sum3-define N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

   5. distribute-lft1-in N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

   7. mul0-lft N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

   8. sum3-undefine N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

   9. +-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right)\right) \]

7. Simplified 78.6%

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

   1. associate-/l/ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{x \cdot x}{s \cdot s}\right), 4\right)\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(x \cdot \frac{x}{s \cdot s}\right), 4\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{x}{s \cdot s}\right)\right), 4\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \left(s \cdot s\right)\right)\right), 4\right)\right)\right) \]

   5. *-lowering-*.f32 85.8%

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(x, \mathsf{*.f32}\left(s, s\right)\right)\right), 4\right)\right)\right) \]

9. Applied egg-rr 85.8%

   \[\leadsto \frac{1}{s \cdot \left(\color{blue}{x \cdot \frac{x}{s \cdot s}} + 4\right)} \]
10. Add Preprocessing

Alternative 9: 46.0% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999987376214e-7) (/ 0.25 s) (/ 1.0 (/ x (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999987376214e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (x / (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.999999987376214e-7) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (x / (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.999999987376214e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x / Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.999999987376214e-7))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (x / (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999e-7

   1. Initial program 99.1%

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

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 38.3%

         \[\leadsto \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right) \]

   7. Simplified 38.3%

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

   if 4.99999999e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(s \cdot \left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right) + \color{blue}{4}\right)\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right) + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + 4\right)\right)\right) \]

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right)\right) \]

   7. Simplified 88.2%

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

   8. Taylor expanded in s around 0

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

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

         \[\leadsto \mathsf{/.f32}\left(s, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(s, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f32 78.3%

         \[\leadsto \mathsf{/.f32}\left(s, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

   10. Simplified 78.3%

       \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
   11. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-*l/ N/A

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

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

          \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{x}{s} \cdot x\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \left(x \cdot \color{blue}{\frac{x}{s}}\right)\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{/.f32}\left(1, \left(x \cdot \frac{1}{\color{blue}{\frac{s}{x}}}\right)\right) \]

       6. un-div-inv N/A

          \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{x}{\color{blue}{\frac{s}{x}}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(x, \color{blue}{\left(\frac{s}{x}\right)}\right)\right) \]

       8. /-lowering-/.f32 79.5%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(x, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right)\right) \]

   12. Applied egg-rr 79.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{s}{x}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 45.3% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999987376214e-7) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999987376214e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = s / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.999999987376214e-7) then
        tmp = 0.25e0 / s
    else
        tmp = s / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.999999987376214e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.999999987376214e-7))
		tmp = single(0.25) / s;
	else
		tmp = s / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999e-7

   1. Initial program 99.1%

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

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in s around inf

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

      1. /-lowering-/.f32 38.3%

         \[\leadsto \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right) \]

   7. Simplified 38.3%

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

   if 4.99999999e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(s \cdot \left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \color{blue}{\left(4 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)\right)}\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right) + \color{blue}{4}\right)\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right) + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + 4\right)\right)\right) \]

      4. sum3-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right), \color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)}, 4\right)\right)\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}\right), \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(\left(0 \cdot \frac{\left|x\right|}{s}\right), \left(\frac{{\color{blue}{\left(\left|x\right|\right)}}^{2}}{{s}^{2}}\right), 4\right)\right)\right) \]

      7. mul0-lft N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{sum3}\left(0, \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{{s}^{2}}\right), 4\right)\right)\right) \]

      8. sum3-undefine N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\left(0 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right) + \color{blue}{4}\right)\right)\right) \]

      9. +-lft-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right), \color{blue}{4}\right)\right)\right) \]

   7. Simplified 88.2%

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

   8. Taylor expanded in s around 0

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

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

         \[\leadsto \mathsf{/.f32}\left(s, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(s, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f32 78.3%

         \[\leadsto \mathsf{/.f32}\left(s, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

   10. Simplified 78.3%

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

Alternative 11: 27.9% accurate, 206.7× 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 99.3%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{0 - \left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in s around inf

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

   1. /-lowering-/.f32 29.2%

      \[\leadsto \mathsf{/.f32}\left(\frac{1}{4}, \color{blue}{s}\right) \]

7. Simplified 29.2%

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

Reproduce

herbie shell --seed 2024288 
(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))))))