Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 13.3s

Alternatives: 16

Speedup: 3.0×

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, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{0 - \frac{\left|x\right|}{s}}\\ \frac{\frac{-1}{\left(\frac{-1}{t\_0} - 2\right) - t\_0}}{s} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (- 0.0 (/ (fabs x) s)))))
   (/ (/ -1.0 (- (- (/ -1.0 t_0) 2.0) t_0)) s)))

C

float code(float x, float s) {
	float t_0 = expf((0.0f - (fabsf(x) / s)));
	return (-1.0f / (((-1.0f / t_0) - 2.0f) - t_0)) / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((0.0e0 - (abs(x) / s)))
    code = ((-1.0e0) / ((((-1.0e0) / t_0) - 2.0e0) - t_0)) / s
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(0.0) - Float32(abs(x) / s)))
	return Float32(Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(-1.0) / t_0) - Float32(2.0)) - t_0)) / s)
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((single(0.0) - (abs(x) / s)));
	tmp = (single(-1.0) / (((single(-1.0) / t_0) - single(2.0)) - t_0)) / s;
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.9%

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

   1. frac-2neg N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{+.f32}\left(\left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{\mathsf{neg}\left(s\right)}}\right), 2\right)\right)\right), s\right) \]

   2. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{+.f32}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}\right)}\right), 2\right)\right)\right), s\right) \]

   3. rec-exp N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{+.f32}\left(\left(\frac{1}{e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}}\right), 2\right)\right)\right), s\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \left(e^{\frac{\mathsf{neg}\left(\left|x\right|\right)}{s}}\right)\right), 2\right)\right)\right), s\right) \]

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

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

   6. distribute-frac-neg N/A

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

   7. distribute-frac-neg2 N/A

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

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\left(\left|x\right|\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), 2\right)\right)\right), s\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), 2\right)\right)\right), s\right) \]

   10. neg-lowering-neg.f32 99.9%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{fabs.f32}\left(x\right), \mathsf{neg.f32}\left(s\right)\right)\right)\right), 2\right)\right)\right), s\right) \]

6. Applied egg-rr 99.9%

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

   1. sub0-neg N/A

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

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

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

   3. fabs-lowering-fabs.f32 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

   \[\leadsto \frac{\frac{-1}{\left(\frac{-1}{e^{0 - \frac{\left|x\right|}{s}}} - 2\right) - e^{0 - \frac{\left|x\right|}{s}}}}{s} \]
10. Add Preprocessing

Alternative 2: 99.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.9%

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

   1. associate-/l/ N/A

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

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

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

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

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

   4. associate-+r+ N/A

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

   5. +-commutative N/A

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

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

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

   7. neg-sub0 N/A

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

   8. frac-2neg N/A

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

   9. distribute-frac-neg2 N/A

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

   10. cosh-undef N/A

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

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

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

   12. distribute-frac-neg N/A

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

   13. cosh-neg N/A

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

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

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

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

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

   16. fabs-lowering-fabs.f32 99.9%

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

6. Applied egg-rr 99.9%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

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

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

   11. fabs-lowering-fabs.f32 99.9%

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

8. Applied egg-rr 99.9%

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

   1. associate-/r* N/A

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

   2. *-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. associate-/r* N/A

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

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

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

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

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

   7. metadata-eval N/A

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

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

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

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

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

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

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

   11. fabs-lowering-fabs.f32 99.9%

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

10. Applied egg-rr 99.9%

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

Alternative 3: 99.6% accurate, 3.0× speedup

Math

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

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(0.5) / Float32(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 99.7%

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

3. Simplified 99.9%

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

   1. associate-/l/ N/A

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

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

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

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

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

   4. associate-+r+ N/A

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

   5. +-commutative N/A

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

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

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

   7. neg-sub0 N/A

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

   8. frac-2neg N/A

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

   9. distribute-frac-neg2 N/A

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

   10. cosh-undef N/A

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

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

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

   12. distribute-frac-neg N/A

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

   13. cosh-neg N/A

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

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

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

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

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

   16. fabs-lowering-fabs.f32 99.9%

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

6. Applied egg-rr 99.9%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

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

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

   11. fabs-lowering-fabs.f32 99.9%

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

8. Applied egg-rr 99.9%

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

   1. associate-/r* N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-rgt-identity N/A

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

   5. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

   10. fabs-lowering-fabs.f32 99.9%

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

10. Applied egg-rr 99.9%

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

Alternative 4: 94.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{0 - \frac{\left|x\right|}{s}}}{s \cdot 4} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (exp (- 0.0 (/ (fabs x) s))) (* s 4.0)))

C

float code(float x, float s) {
	return expf((0.0f - (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((0.0e0 - (abs(x) / s))) / (s * 4.0e0)
end function

Julia

function code(x, s)
	return Float32(exp(Float32(Float32(0.0) - Float32(abs(x) / s))) / Float32(s * Float32(4.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = exp((single(0.0) - (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

   \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \color{blue}{\left(4 \cdot s\right)}\right) \]
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 95.9%

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

5. Simplified 95.9%

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

6. Final simplification 95.9%

   \[\leadsto \frac{e^{0 - \frac{\left|x\right|}{s}}}{s \cdot 4} \]
7. Add Preprocessing

Alternative 5: 94.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = (single(0.25) / s) / exp((abs(x) / 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. 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 95.9%

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

5. Simplified 95.9%

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

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-frac-neg2 N/A

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

   5. associate-/r* N/A

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

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

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

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

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

   10. metadata-eval N/A

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

   11. rec-exp N/A

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

   12. distribute-frac-neg2 N/A

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

   13. remove-double-neg N/A

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

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

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

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

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

   16. fabs-lowering-fabs.f32 95.9%

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

7. Applied egg-rr 95.9%

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

Alternative 6: 74.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s} + \frac{s}{0.25}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= s 1.5000000786160286e-23)
   (/ (/ 1.0 s) (/ (* x x) (* s s)))
   (/ 1.0 (+ (* x (/ x s)) (/ s 0.25)))))

C

float code(float x, float s) {
	float tmp;
	if (s <= 1.5000000786160286e-23f) {
		tmp = (1.0f / s) / ((x * x) / (s * s));
	} else {
		tmp = 1.0f / ((x * (x / s)) + (s / 0.25f));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (s <= 1.5000000786160286e-23) then
        tmp = (1.0e0 / s) / ((x * x) / (s * s))
    else
        tmp = 1.0e0 / ((x * (x / s)) + (s / 0.25e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (s <= Float32(1.5000000786160286e-23))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(x * x) / Float32(s * s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(x / s)) + Float32(s / Float32(0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (s <= single(1.5000000786160286e-23))
		tmp = (single(1.0) / s) / ((x * x) / (s * s));
	else
		tmp = single(1.0) / ((x * (x / s)) + (s / single(0.25)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 1.50000008e-23

   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%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

      9. mul0-lft N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

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

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

   7. Simplified 76.9%

      \[\leadsto \frac{\frac{1}{\color{blue}{4 - \frac{0 - \frac{x \cdot x}{s}}{s}}}}{s} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(4 - \frac{0 - \frac{x \cdot x}{s}}{s}\right) \cdot s}} \]

      2. associate-/l/ N/A

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{4 - \frac{0 - \frac{x \cdot x}{s}}{s}}} \]

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

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

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

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

      5. div-inv N/A

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

      6. sub0-neg N/A

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

      7. cancel-sign-sub N/A

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

      8. div-inv N/A

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

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

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

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

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

      11. associate-/l* N/A

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

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

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

      13. /-lowering-/.f32 77.3%

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

   9. Applied egg-rr 77.3%

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

   10. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f32 84.7%

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

   12. Simplified 84.7%

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

   if 1.50000008e-23 < s

   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 99.9%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

      9. mul0-lft N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

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

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

   7. Simplified 76.0%

      \[\leadsto \frac{\frac{1}{\color{blue}{4 - \frac{0 - \frac{x \cdot x}{s}}{s}}}}{s} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{s \cdot \left(4 - \frac{0 - \frac{x \cdot x}{s}}{s}\right)}} \]

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

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

      3. div-inv N/A

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

      4. sub0-neg N/A

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

      5. cancel-sign-sub N/A

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

      6. div-inv N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. associate-/r/ N/A

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

      11. clear-num N/A

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

      12. div-inv N/A

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

      13. associate-*l* N/A

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

   9. Applied egg-rr 71.4%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \frac{x}{s} + \frac{s}{0.25}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.7% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s} + \frac{s}{0.25}}\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \frac{s}{x \cdot x}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 5.00000006675716e-11)
   (/ 1.0 (+ (* x (/ x s)) (/ s 0.25)))
   (/ (* s (/ s (* x x))) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 5.00000006675716e-11f) {
		tmp = 1.0f / ((x * (x / s)) + (s / 0.25f));
	} else {
		tmp = (s * (s / (x * x))) / 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 <= 5.00000006675716e-11) then
        tmp = 1.0e0 / ((x * (x / s)) + (s / 0.25e0))
    else
        tmp = (s * (s / (x * x))) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(5.00000006675716e-11))
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(x / s)) + Float32(s / Float32(0.25))));
	else
		tmp = Float32(Float32(s * Float32(s / Float32(x * x))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(5.00000006675716e-11))
		tmp = single(1.0) / ((x * (x / s)) + (s / single(0.25)));
	else
		tmp = (s * (s / (x * x))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5.00000007e-11

   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 99.9%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

      9. mul0-lft N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

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

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

   7. Simplified 73.8%

      \[\leadsto \frac{\frac{1}{\color{blue}{4 - \frac{0 - \frac{x \cdot x}{s}}{s}}}}{s} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{s \cdot \left(4 - \frac{0 - \frac{x \cdot x}{s}}{s}\right)}} \]

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

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

      3. div-inv N/A

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

      4. sub0-neg N/A

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

      5. cancel-sign-sub N/A

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

      6. div-inv N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. associate-/r/ N/A

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

      11. clear-num N/A

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

      12. div-inv N/A

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

      13. associate-*l* N/A

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

   9. Applied egg-rr 66.0%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \frac{x}{s} + \frac{s}{0.25}}} \]

   if 5.00000007e-11 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

      9. mul0-lft N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

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

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

   7. Simplified 81.3%

      \[\leadsto \frac{\frac{1}{\color{blue}{4 - \frac{0 - \frac{x \cdot x}{s}}{s}}}}{s} \]

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f32 74.6%

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

   10. Simplified 74.6%

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

Alternative 8: 81.4% accurate, 41.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.9%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

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

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

   4. associate-+r+ N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. distribute-rgt1-in N/A

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

   8. metadata-eval N/A

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

   9. mul0-lft N/A

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

   10. neg-sub0 N/A

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

   11. mul-1-neg N/A

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

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

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

7. Simplified 76.4%

   \[\leadsto \frac{\frac{1}{\color{blue}{4 - \frac{0 - \frac{x \cdot x}{s}}{s}}}}{s} \]
8. Step-by-step derivation

   1. sub0-neg N/A

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

   2. associate-/l* N/A

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

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

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

   4. associate-/l* N/A

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

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

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

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

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

   7. associate-/l/ N/A

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

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

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

   9. *-lowering-*.f32 82.3%

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

9. Applied egg-rr 82.3%

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

    1. clear-num N/A

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

    2. associate-/r/ N/A

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

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

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

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

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

    5. *-lowering-*.f32 82.7%

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

11. Applied egg-rr 82.7%

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

12. Final simplification 82.7%

    \[\leadsto \frac{\frac{-1}{x \cdot \left(x \cdot \frac{-1}{s \cdot s}\right) - 4}}{s} \]
13. Add Preprocessing

Alternative 9: 49.2% accurate, 44.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \frac{s}{x \cdot x}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999920083944e-12) (/ 0.25 s) (/ (* s (/ s (* x x))) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999920083944e-12f) {
		tmp = 0.25f / s;
	} else {
		tmp = (s * (s / (x * x))) / 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 <= 1.9999999920083944e-12) then
        tmp = 0.25e0 / s
    else
        tmp = (s * (s / (x * x))) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999920083944e-12))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(s * Float32(s / Float32(x * x))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999920083944e-12))
		tmp = single(0.25) / s;
	else
		tmp = (s * (s / (x * x))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999e-12

   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

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

      1. /-lowering-/.f32 37.5%

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

   5. Simplified 37.5%

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

   if 1.99999999e-12 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

      9. mul0-lft N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

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

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

   7. Simplified 81.3%

      \[\leadsto \frac{\frac{1}{\color{blue}{4 - \frac{0 - \frac{x \cdot x}{s}}{s}}}}{s} \]

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f32 74.6%

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

   10. Simplified 74.6%

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

Alternative 10: 81.7% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.9%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

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

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

   4. associate-+r+ N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. distribute-rgt1-in N/A

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

   8. metadata-eval N/A

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

   9. mul0-lft N/A

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

   10. neg-sub0 N/A

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

   11. mul-1-neg N/A

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

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

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

7. Simplified 76.4%

   \[\leadsto \frac{\frac{1}{\color{blue}{4 - \frac{0 - \frac{x \cdot x}{s}}{s}}}}{s} \]
8. Step-by-step derivation

   1. sub0-neg N/A

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

   2. associate-/l* N/A

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

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

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

   4. associate-/l* N/A

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

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

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

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

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

   7. associate-/l/ N/A

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

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

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

   9. *-lowering-*.f32 82.3%

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

9. Applied egg-rr 82.3%

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

10. Final simplification 82.3%

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

Alternative 11: 81.5% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.9%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

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

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

   4. associate-+r+ N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. distribute-rgt1-in N/A

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

   8. metadata-eval N/A

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

   9. mul0-lft N/A

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

   10. neg-sub0 N/A

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

   11. mul-1-neg N/A

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

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

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

7. Simplified 76.4%

   \[\leadsto \frac{\frac{1}{\color{blue}{4 - \frac{0 - \frac{x \cdot x}{s}}{s}}}}{s} \]
8. Step-by-step derivation

   1. sub0-neg N/A

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

   2. associate-/l* N/A

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

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

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

   4. associate-/l* N/A

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

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

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

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

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

   7. associate-/l/ N/A

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

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

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

   9. *-lowering-*.f32 82.3%

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

9. Applied egg-rr 82.3%

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

    1. associate-/r* N/A

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

    2. associate-/l/ N/A

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

    3. cancel-sign-sub N/A

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

    4. +-commutative N/A

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

    5. div-inv N/A

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

    6. associate-*r* N/A

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

    7. metadata-eval N/A

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

    8. lft-mult-inverse N/A

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

    9. associate-*l* N/A

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

    10. div-inv N/A

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

    11. *-commutative N/A

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

    12. distribute-lft-in N/A

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

    13. associate-*l* N/A

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

    14. associate-*l* N/A

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

    15. associate-/r* N/A

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

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

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

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

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

11. Applied egg-rr 82.3%

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

Alternative 12: 45.7% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999858590343e-10) (/ 0.25 s) (/ 1.0 (/ (* x x) s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999858590343e-10f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / ((x * x) / 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 <= 4.999999858590343e-10) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / ((x * x) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.999999858590343e-10))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * x) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.999999858590343e-10))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / ((x * x) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999986e-10

   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

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

      1. /-lowering-/.f32 37.4%

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

   5. Simplified 37.4%

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

   if 4.99999986e-10 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

      9. mul0-lft N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

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

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

   7. Simplified 82.1%

      \[\leadsto \frac{\frac{1}{\color{blue}{4 - \frac{0 - \frac{x \cdot x}{s}}{s}}}}{s} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   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 64.3%

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

   10. Simplified 64.3%

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

       1. associate-/r* N/A

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

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

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

       3. /-lowering-/.f32 64.3%

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

   12. Applied egg-rr 64.3%

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

       1. clear-num N/A

          \[\leadsto \frac{\frac{1}{\frac{x}{s}}}{x} \]

       2. associate-/l/ N/A

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

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

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

       4. associate-*r/ N/A

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

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

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

       6. *-lowering-*.f32 67.1%

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

   14. Applied egg-rr 67.1%

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

Alternative 13: 45.7% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999858590343e-10) (/ 0.25 s) (/ 1.0 (/ x (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999858590343e-10f) {
		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 <= 4.999999858590343e-10) 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(4.999999858590343e-10))
		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(4.999999858590343e-10))
		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 < 4.99999986e-10

   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

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

      1. /-lowering-/.f32 37.4%

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

   5. Simplified 37.4%

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

   if 4.99999986e-10 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

      9. mul0-lft N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

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

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

   7. Simplified 82.1%

      \[\leadsto \frac{\frac{1}{\color{blue}{4 - \frac{0 - \frac{x \cdot x}{s}}{s}}}}{s} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   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 64.3%

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

   10. Simplified 64.3%

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

       1. clear-num N/A

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

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

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

       3. clear-num N/A

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

       4. associate-/r* N/A

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

       5. clear-num N/A

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

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

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

       7. /-lowering-/.f32 67.1%

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

   12. Applied egg-rr 67.1%

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

Alternative 14: 45.0% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999858590343e-10) (/ 0.25 s) (/ (/ s x) x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999858590343e-10f) {
		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 <= 4.999999858590343e-10) 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(4.999999858590343e-10))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(s / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.999999858590343e-10))
		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 < 4.99999986e-10

   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

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

      1. /-lowering-/.f32 37.4%

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

   5. Simplified 37.4%

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

   if 4.99999986e-10 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

      9. mul0-lft N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

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

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

   7. Simplified 82.1%

      \[\leadsto \frac{\frac{1}{\color{blue}{4 - \frac{0 - \frac{x \cdot x}{s}}{s}}}}{s} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   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 64.3%

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

   10. Simplified 64.3%

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

       1. associate-/r* N/A

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

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

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

       3. /-lowering-/.f32 64.3%

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

   12. Applied egg-rr 64.3%

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

Alternative 15: 45.1% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999858590343e-10) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999858590343e-10f) {
		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 <= 4.999999858590343e-10) 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(4.999999858590343e-10))
		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(4.999999858590343e-10))
		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 < 4.99999986e-10

   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

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

      1. /-lowering-/.f32 37.4%

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

   5. Simplified 37.4%

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

   if 4.99999986e-10 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. associate-+r+ N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

      9. mul0-lft N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

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

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

   7. Simplified 82.1%

      \[\leadsto \frac{\frac{1}{\color{blue}{4 - \frac{0 - \frac{x \cdot x}{s}}{s}}}}{s} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   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 64.3%

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

   10. Simplified 64.3%

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

Alternative 16: 26.9% 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 99.7%

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

3. Taylor expanded in s around inf

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

   1. /-lowering-/.f32 26.6%

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

5. Simplified 26.6%

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

Reproduce

herbie shell --seed 2024164 
(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))))))