Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 13.0s

Alternatives: 17

Speedup: 2.9×

Specification

\[0 \leq s \land s \leq 1.0651631\]

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{e^{\frac{0 - \left|x\right|}{s}} + \left(2 + {\left({e}^{\left(\frac{\frac{\left|x\right|}{s}}{2}\right)}\right)}^{2}\right)}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  (/
   1.0
   (+
    (exp (/ (- 0.0 (fabs x)) s))
    (+ 2.0 (pow (pow E (/ (/ (fabs x) s) 2.0)) 2.0))))
  s))

C

float code(float x, float s) {
	return (1.0f / (expf(((0.0f - fabsf(x)) / s)) + (2.0f + powf(powf(((float) M_E), ((fabsf(x) / s) / 2.0f)), 2.0f)))) / s;
}

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(Float32(0.0) - abs(x)) / s)) + Float32(Float32(2.0) + ((Float32(exp(1)) ^ Float32(Float32(abs(x) / s) / Float32(2.0))) ^ Float32(2.0))))) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / (exp(((single(0.0) - abs(x)) / s)) + (single(2.0) + ((single(2.71828182845904523536) ^ ((abs(x) / s) / single(2.0))) ^ single(2.0))))) / s;
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. *-lft-identity N/A

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

   2. exp-prod N/A

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

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

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

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

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

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

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

   6. fabs-lowering-fabs.f32 99.9%

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

6. Applied egg-rr 99.9%

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

   1. metadata-eval N/A

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

   2. pow-exp N/A

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

   3. pow-unpow N/A

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

   4. neg-mul-1 N/A

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

   5. distribute-frac-neg2 N/A

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

   6. sqr-pow N/A

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

   7. pow2 N/A

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

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

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

   9. distribute-frac-neg2 N/A

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

   10. neg-mul-1 N/A

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

   11. associate-/l* N/A

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

   12. pow-unpow N/A

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

   13. pow-exp N/A

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

   14. metadata-eval N/A

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

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

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

   16. exp-1-e N/A

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

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

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

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

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

   19. /-lowering-/.f32 N/A

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

   20. fabs-lowering-fabs.f32 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = (0.0f - fabsf(x)) / s;
	return (1.0f / (expf(t_0) + (2.0f + powf(expf(-1.0f), 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 = (0.0e0 - abs(x)) / s
    code = (1.0e0 / (exp(t_0) + (2.0e0 + (exp((-1.0e0)) ** t_0)))) / s
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. remove-double-div N/A

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

   2. exp-neg N/A

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

   3. neg-mul-1 N/A

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

   4. exp-prod N/A

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

   5. pow-flip N/A

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

   6. distribute-frac-neg N/A

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

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

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

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

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

   9. distribute-frac-neg N/A

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

   10. distribute-neg-frac2 N/A

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

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

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

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

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

   13. neg-lowering-neg.f32 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 3: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{e^{\frac{0 - \left|x\right|}{s}} + \left(2 + {e}^{\left(\frac{\left|x\right|}{s}\right)}\right)}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (/ 1.0 (+ (exp (/ (- 0.0 (fabs x)) s)) (+ 2.0 (pow E (/ (fabs x) s))))) s))

C

float code(float x, float s) {
	return (1.0f / (expf(((0.0f - fabsf(x)) / s)) + (2.0f + powf(((float) M_E), (fabsf(x) / s))))) / s;
}

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(Float32(0.0) - abs(x)) / s)) + Float32(Float32(2.0) + (Float32(exp(1)) ^ Float32(abs(x) / s))))) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / (exp(((single(0.0) - abs(x)) / s)) + (single(2.0) + (single(2.71828182845904523536) ^ (abs(x) / s))))) / s;
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. *-lft-identity N/A

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

   2. exp-prod N/A

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

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

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

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

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

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

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

   6. fabs-lowering-fabs.f32 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 4: 99.6% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (/ (exp (- (log (+ 2.0 (* 2.0 (cosh (/ (fabs x) s))))))) s))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. inv-pow N/A

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

   2. pow-to-exp N/A

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

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

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

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

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

6. Applied egg-rr 99.8%

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

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

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

   2. *-commutative N/A

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

   3. mul-1-neg N/A

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

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

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

   5. rem-exp-log N/A

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

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

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

   7. rem-exp-log N/A

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

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

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

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

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

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

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

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

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

   12. fabs-lowering-fabs.f32 99.8%

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

8. Applied egg-rr 99.8%

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

Alternative 5: 99.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. associate-/l/ N/A

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

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

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

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

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

   4. associate-+r+ N/A

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

   5. +-commutative N/A

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

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

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

   7. distribute-frac-neg N/A

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

   8. +-commutative N/A

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

   9. distribute-frac-neg N/A

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

   10. cosh-undef N/A

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

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

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

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

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

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

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

   14. fabs-lowering-fabs.f32 99.8%

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

6. Applied egg-rr 99.8%

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

Alternative 6: 94.9% accurate, 3.0× speedup

Math

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

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(Float32(0.0) - 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 99.8%

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

3. Taylor expanded in s around inf

   \[\leadsto \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{neg.f32}\left(\mathsf{fabs.f32}\left(x\right)\right), s\right)\right), \color{blue}{\left(4 \cdot s\right)}\right) \]
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 96.2%

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

5. Simplified 96.2%

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

6. Final simplification 96.2%

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

Alternative 7: 84.0% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.0000000126843074 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{4 + x \cdot \left(\frac{1}{s} \cdot \frac{x}{s}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + x \cdot \left(x \cdot \frac{1}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.0000000126843074e-28)
   (/ (/ 1.0 (+ 4.0 (* x (* (/ 1.0 s) (/ x s))))) s)
   (/ (/ 1.0 s) (+ 4.0 (* x (* x (/ 1.0 (* s s))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.0000000126843074e-28f) {
		tmp = (1.0f / (4.0f + (x * ((1.0f / s) * (x / s))))) / s;
	} else {
		tmp = (1.0f / s) / (4.0f + (x * (x * (1.0f / (s * s)))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.0000000126843074e-28) then
        tmp = (1.0e0 / (4.0e0 + (x * ((1.0e0 / s) * (x / s))))) / s
    else
        tmp = (1.0e0 / s) / (4.0e0 + (x * (x * (1.0e0 / (s * s)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.0000000126843074e-28))
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(4.0) + Float32(x * Float32(Float32(Float32(1.0) / s) * Float32(x / s))))) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(x * Float32(x * Float32(Float32(1.0) / Float32(s * s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.0000000126843074e-28))
		tmp = (single(1.0) / (single(4.0) + (x * ((single(1.0) / s) * (x / s))))) / s;
	else
		tmp = (single(1.0) / s) / (single(4.0) + (x * (x * (single(1.0) / (s * s)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.00000001e-28

   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 99.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{-\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}}{s}} \]
   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 77.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. distribute-neg-frac N/A

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

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

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

      6. distribute-frac-neg2 N/A

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

      7. remove-double-neg N/A

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

      8. associate-/l* N/A

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

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

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

      10. /-lowering-/.f32 78.4%

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

   9. Applied egg-rr 78.4%

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

       1. associate-*l/ N/A

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

       2. div-inv N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

       7. /-lowering-/.f32 86.3%

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

   11. Applied egg-rr 86.3%

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

   if 4.00000001e-28 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{-\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}}{s}} \]
   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 79.1%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. distribute-neg-frac N/A

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

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

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

      9. distribute-frac-neg2 N/A

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

      10. remove-double-neg N/A

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

      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 79.1%

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

   9. Applied egg-rr 79.1%

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

       1. associate-*r/ N/A

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

       2. associate-/r* N/A

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

       3. clear-num N/A

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

       4. associate-/r/ N/A

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

       5. associate-*r* N/A

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

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

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

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

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

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

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

       9. *-lowering-*.f32 91.7%

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

   11. Applied egg-rr 91.7%

       \[\leadsto \frac{\frac{1}{s}}{4 + \color{blue}{\left(\frac{1}{s \cdot s} \cdot x\right) \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.0000000126843074 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{4 + x \cdot \left(\frac{1}{s} \cdot \frac{x}{s}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + x \cdot \left(x \cdot \frac{1}{s \cdot s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.5% accurate, 34.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.000000157232057 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x \cdot x}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 3.000000157232057e-23)
   (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (/ x s))))
   (/ 1.0 (* s (+ 4.0 (/ (* x x) (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 3.000000157232057e-23f) {
		tmp = (1.0f / s) / (4.0f + ((x / s) * (x / s)));
	} else {
		tmp = 1.0f / (s * (4.0f + ((x * x) / (s * s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 3.000000157232057e-23) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) * (x / s)))
    else
        tmp = 1.0e0 / (s * (4.0e0 + ((x * x) / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(3.000000157232057e-23))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(Float32(x * x) / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(3.000000157232057e-23))
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * (x / s)));
	else
		tmp = single(1.0) / (s * (single(4.0) + ((x * x) / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.00000016e-23

   1. Initial program 99.7%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{-\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}}{s}} \]
   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 76.8%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. distribute-neg-frac N/A

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

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

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

      9. distribute-frac-neg2 N/A

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

      10. remove-double-neg N/A

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

      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 77.4%

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

   9. Applied egg-rr 77.4%

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

       1. associate-*l/ N/A

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

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

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

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

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

       4. /-lowering-/.f32 77.4%

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

   11. Applied egg-rr 77.4%

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

   if 3.00000016e-23 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

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

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

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

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in s around inf

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

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

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

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

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

      3. unpow2 N/A

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

      4. sqr-abs N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f32 88.3%

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

   9. Simplified 88.3%

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

Alternative 9: 50.4% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.0299999920896155 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{x \cdot x}{s \cdot s}}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.0299999920896155e-13)
   (/ 0.25 s)
   (/ (/ 1.0 (/ (* x x) (* s s))) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.0299999920896155e-13f) {
		tmp = 0.25f / s;
	} else {
		tmp = (1.0f / ((x * x) / (s * s))) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.0299999920896155e-13) then
        tmp = 0.25e0 / s
    else
        tmp = (1.0e0 / ((x * x) / (s * s))) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.0299999920896155e-13))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(x * x) / Float32(s * s))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.0299999920896155e-13))
		tmp = single(0.25) / s;
	else
		tmp = (single(1.0) / ((x * x) / (s * s))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.02999999e-13

   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 39.0%

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

   5. Simplified 39.0%

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

   if 1.02999999e-13 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{-\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}}{s}} \]
   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 85.2%

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

   8. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f32 86.4%

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

   10. Simplified 86.4%

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

Alternative 10: 49.4% accurate, 44.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.0299999920896155 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot s}{x \cdot x}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.0299999920896155e-13) (/ 0.25 s) (/ (/ (* s s) (* x x)) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.0299999920896155e-13f) {
		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 <= 1.0299999920896155e-13) 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(1.0299999920896155e-13))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(s * s) / Float32(x * x)) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.0299999920896155e-13))
		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 < 1.02999999e-13

   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 39.0%

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

   5. Simplified 39.0%

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

   if 1.02999999e-13 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{-\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}}{s}} \]
   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 85.2%

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

   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. /-lowering-/.f32 N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f32 82.4%

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

   10. Simplified 82.4%

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

Alternative 11: 81.6% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{\frac{1}{e^{-\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}}{s}} \]
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 78.4%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

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

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

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

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

   5. sub-neg N/A

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

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

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

   7. distribute-neg-frac N/A

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

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

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

   9. distribute-frac-neg2 N/A

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

   10. remove-double-neg N/A

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

   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 78.7%

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

9. Applied egg-rr 78.7%

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

    1. associate-/l* N/A

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

    2. *-commutative N/A

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

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

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

    4. associate-/r* N/A

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

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

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

    6. *-lowering-*.f32 85.8%

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

11. Applied egg-rr 85.8%

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

12. Final simplification 85.8%

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

Alternative 12: 78.0% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

   3. clear-num N/A

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

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

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

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

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in s around inf

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

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

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

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

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

   3. unpow2 N/A

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

   4. sqr-abs N/A

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f32 80.2%

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

9. Simplified 80.2%

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

Alternative 13: 45.0% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.0000000233721948e-7) (/ 0.25 s) (/ 1.0 (/ x (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.0000000233721948e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (x / (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 2.0000000233721948e-7) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (x / (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.0000000233721948e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x / Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.0000000233721948e-7))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (x / (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.00000002e-7

   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 37.8%

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

   5. Simplified 37.8%

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

   if 2.00000002e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{-\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}}{s}} \]
   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 88.0%

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

   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 73.0%

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

   10. Simplified 73.0%

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

       1. clear-num N/A

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

       2. associate-*r/ N/A

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

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

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

       4. clear-num N/A

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

       5. un-div-inv 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 74.5%

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

   12. Applied egg-rr 74.5%

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

Alternative 14: 45.0% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.0000000233721948e-7) (/ 0.25 s) (/ 1.0 (* x (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.0000000233721948e-7f) {
		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 <= 2.0000000233721948e-7) 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(2.0000000233721948e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.0000000233721948e-7))
		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 < 2.00000002e-7

   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 37.8%

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

   5. Simplified 37.8%

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

   if 2.00000002e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{-\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}}{s}} \]
   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 88.0%

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

   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 73.0%

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

   10. Simplified 73.0%

       \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
   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 73.0%

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

   12. Applied egg-rr 73.0%

       \[\leadsto \color{blue}{\frac{\frac{s}{x}}{x}} \]
   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. *-lowering-*.f32 N/A

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

       5. /-lowering-/.f32 74.5%

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

   14. Applied egg-rr 74.5%

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

Alternative 15: 44.3% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.0000000233721948e-7) (/ 0.25 s) (* s (/ 1.0 (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.0000000233721948e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = s * (1.0f / (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 <= 2.0000000233721948e-7) then
        tmp = 0.25e0 / s
    else
        tmp = s * (1.0e0 / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.0000000233721948e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s * Float32(Float32(1.0) / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.0000000233721948e-7))
		tmp = single(0.25) / s;
	else
		tmp = s * (single(1.0) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.00000002e-7

   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 37.8%

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

   5. Simplified 37.8%

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

   if 2.00000002e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{-\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}}{s}} \]
   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 88.0%

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

   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 73.0%

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

   10. Simplified 73.0%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

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

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

       5. *-lowering-*.f32 73.0%

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

   12. Applied egg-rr 73.0%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{1}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.3% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.0000000233721948e-7) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.0000000233721948e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = s / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 2.0000000233721948e-7) then
        tmp = 0.25e0 / s
    else
        tmp = s / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.0000000233721948e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.0000000233721948e-7))
		tmp = single(0.25) / s;
	else
		tmp = s / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.00000002e-7

   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 37.8%

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

   5. Simplified 37.8%

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

   if 2.00000002e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{e^{-\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}}{s}} \]
   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 88.0%

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

   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 73.0%

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

   10. Simplified 73.0%

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

Alternative 17: 26.7% 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 99.8%

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

3. Taylor expanded in s around inf

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

   1. /-lowering-/.f32 27.9%

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

5. Simplified 27.9%

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

Reproduce

herbie shell --seed 2024155 
(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))))))