Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 17.4s

Alternatives: 17

Speedup: 3.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

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. Step-by-step derivation

   1. distribute-rgt-in N/A

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

   2. +-commutative N/A

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

   3. associate-*l* N/A

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

   4. distribute-lft-out N/A

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

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

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

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

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

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. Step-by-step derivation

   1. distribute-rgt-in N/A

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

   2. +-commutative N/A

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

   3. associate-*l* N/A

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

   4. distribute-lft-out N/A

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

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

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

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

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

8. Applied egg-rr 99.8%

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

   1. associate-/r* N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. distribute-lft1-in N/A

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

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

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

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

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

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

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

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

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

   10. fabs-lowering-fabs.f32 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 3: 95.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 94.5%

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

5. Simplified 94.5%

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

6. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

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

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

   3. mul-1-neg N/A

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

   4. rec-exp N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

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

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

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

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

   10. fabs-lowering-fabs.f32 94.5%

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

8. Simplified 94.5%

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

Alternative 4: 87.0% accurate, 29.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot x}{s}\\ \frac{\frac{-4}{\frac{t\_0}{s \cdot \frac{s}{t\_0}} + -16}}{s} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (* x x) s))) (/ (/ -4.0 (+ (/ t_0 (* s (/ s t_0))) -16.0)) s)))

C

float code(float x, float s) {
	float t_0 = (x * x) / s;
	return (-4.0f / ((t_0 / (s * (s / t_0))) + -16.0f)) / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = (x * x) / s
    code = ((-4.0e0) / ((t_0 / (s * (s / t_0))) + (-16.0e0))) / s
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(x * x) / s)
	return Float32(Float32(Float32(-4.0) / Float32(Float32(t_0 / Float32(s * Float32(s / t_0))) + Float32(-16.0))) / s)
end

MATLAB

function tmp = code(x, s)
	t_0 = (x * x) / s;
	tmp = (single(-4.0) / ((t_0 / (s * (s / t_0))) + single(-16.0))) / s;
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. mul0-lft N/A

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

   6. +-lft-identity N/A

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

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

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

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

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

   9. unpow2 N/A

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

   10. sqr-abs N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f32 81.8%

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

7. Simplified 81.8%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

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

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

   4. sub-neg N/A

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

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

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

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

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

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

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

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

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

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

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

9. Applied egg-rr 28.8%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\color{blue}{-4}, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), s\right), \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, x\right), s\right)\right)\right)\right), -16\right)\right), s\right) \]
11. Step-by-step derivation

12. Simplified 87.6%

    \[\leadsto \frac{\frac{\color{blue}{-4}}{\frac{\frac{x \cdot x}{s}}{s \cdot \frac{s}{\frac{x \cdot x}{s}}} + -16}}{s} \]
13. Add Preprocessing

Alternative 5: 80.6% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{1}{\frac{x}{s} \cdot \frac{x}{s} + 4}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4 - x \cdot \left(x \cdot \frac{-1}{s \cdot s}\right)}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999593223797e-31)
   (/ (/ 1.0 (+ (* (/ x s) (/ x s)) 4.0)) s)
   (/ (/ 1.0 (- 4.0 (* x (* x (/ -1.0 (* s s)))))) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999593223797e-31f) {
		tmp = (1.0f / (((x / s) * (x / s)) + 4.0f)) / s;
	} else {
		tmp = (1.0f / (4.0f - (x * (x * (-1.0f / (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 <= 1.9999999593223797e-31) then
        tmp = (1.0e0 / (((x / s) * (x / s)) + 4.0e0)) / s
    else
        tmp = (1.0e0 / (4.0e0 - (x * (x * ((-1.0e0) / (s * s)))))) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999593223797e-31))
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(Float32(x / s) * Float32(x / s)) + Float32(4.0))) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(4.0) - Float32(x * Float32(x * Float32(Float32(-1.0) / Float32(s * s)))))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999593223797e-31))
		tmp = (single(1.0) / (((x / s) * (x / s)) + single(4.0))) / s;
	else
		tmp = (single(1.0) / (single(4.0) - (x * (x * (single(-1.0) / (s * s)))))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999996e-31

   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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 80.0%

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

   7. Simplified 80.0%

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

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

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

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

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

      4. /-lowering-/.f32 79.5%

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

   9. Applied egg-rr 79.5%

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

   if 1.99999996e-31 < x

   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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 84.1%

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

   7. Simplified 84.1%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

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

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. remove-double-neg N/A

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

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

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

      12. *-lowering-*.f32 86.6%

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

   9. Applied egg-rr 86.6%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{1}{\frac{x}{s} \cdot \frac{x}{s} + 4}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4 - x \cdot \left(x \cdot \frac{-1}{s \cdot s}\right)}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 34.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{1}{\frac{x}{s} \cdot \frac{x}{s} + 4}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4 + x \cdot \frac{x}{s \cdot s}}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999593223797e-31)
   (/ (/ 1.0 (+ (* (/ x s) (/ x s)) 4.0)) s)
   (/ (/ 1.0 (+ 4.0 (* x (/ x (* s s))))) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999593223797e-31f) {
		tmp = (1.0f / (((x / s) * (x / s)) + 4.0f)) / s;
	} else {
		tmp = (1.0f / (4.0f + (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 <= 1.9999999593223797e-31) then
        tmp = (1.0e0 / (((x / s) * (x / s)) + 4.0e0)) / s
    else
        tmp = (1.0e0 / (4.0e0 + (x * (x / (s * s))))) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999593223797e-31))
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(Float32(x / s) * Float32(x / s)) + Float32(4.0))) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(4.0) + Float32(x * Float32(x / Float32(s * s))))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999593223797e-31))
		tmp = (single(1.0) / (((x / s) * (x / s)) + single(4.0))) / s;
	else
		tmp = (single(1.0) / (single(4.0) + (x * (x / (s * s))))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999996e-31

   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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 80.0%

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

   7. Simplified 80.0%

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

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

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

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

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

      4. /-lowering-/.f32 79.5%

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

   9. Applied egg-rr 79.5%

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

   if 1.99999996e-31 < x

   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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 84.1%

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

   7. Simplified 84.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. associate-/r* N/A

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

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

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

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

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

      7. *-lowering-*.f32 84.0%

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

   9. Applied egg-rr 84.0%

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

   10. Taylor expanded in s around 0

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

       1. unpow2 N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f32 86.7%

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

   12. Simplified 86.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{1}{\frac{x}{s} \cdot \frac{x}{s} + 4}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4 + x \cdot \frac{x}{s \cdot s}}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.9% accurate, 34.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}} + s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4 + x \cdot \frac{x}{s \cdot s}}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999593223797e-31)
   (/ 1.0 (+ (/ x (/ s x)) (* s 4.0)))
   (/ (/ 1.0 (+ 4.0 (* x (/ x (* s s))))) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999593223797e-31f) {
		tmp = 1.0f / ((x / (s / x)) + (s * 4.0f));
	} else {
		tmp = (1.0f / (4.0f + (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 <= 1.9999999593223797e-31) then
        tmp = 1.0e0 / ((x / (s / x)) + (s * 4.0e0))
    else
        tmp = (1.0e0 / (4.0e0 + (x * (x / (s * s))))) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999593223797e-31))
		tmp = Float32(Float32(1.0) / Float32(Float32(x / Float32(s / x)) + Float32(s * Float32(4.0))));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(4.0) + Float32(x * Float32(x / Float32(s * s))))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999593223797e-31))
		tmp = single(1.0) / ((x / (s / x)) + (s * single(4.0)));
	else
		tmp = (single(1.0) / (single(4.0) + (x * (x / (s * s))))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999996e-31

   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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 80.0%

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

   7. Simplified 80.0%

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

      1. clear-num N/A

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

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

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

      3. associate-/r/ N/A

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

      4. /-rgt-identity N/A

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f32 79.9%

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

   9. Applied egg-rr 79.9%

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

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

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

       2. distribute-rgt-in N/A

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

       3. div-inv N/A

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

       4. associate-*l* N/A

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

       5. pow2 N/A

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

       6. pow-flip N/A

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

       7. pow-plus N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. inv-pow N/A

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

       11. div-inv N/A

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

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

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

       13. clear-num N/A

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

       14. associate-/l/ N/A

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

       15. clear-num N/A

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

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

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

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

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

       18. *-commutative N/A

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

       19. *-lowering-*.f32 71.9%

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

   11. Applied egg-rr 71.9%

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

   if 1.99999996e-31 < x

   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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 84.1%

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

   7. Simplified 84.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. associate-/r* N/A

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

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

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

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

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

      7. *-lowering-*.f32 84.0%

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

   9. Applied egg-rr 84.0%

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

   10. Taylor expanded in s around 0

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

       1. unpow2 N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f32 86.7%

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

   12. Simplified 86.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}} + s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4 + x \cdot \frac{x}{s \cdot s}}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.9% accurate, 34.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}} + s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + x \cdot \frac{x}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999593223797e-31)
   (/ 1.0 (+ (/ x (/ s x)) (* s 4.0)))
   (/ 1.0 (* s (+ 4.0 (* x (/ x (* s s))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999593223797e-31f) {
		tmp = 1.0f / ((x / (s / x)) + (s * 4.0f));
	} 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 <= 1.9999999593223797e-31) then
        tmp = 1.0e0 / ((x / (s / x)) + (s * 4.0e0))
    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(1.9999999593223797e-31))
		tmp = Float32(Float32(1.0) / Float32(Float32(x / Float32(s / x)) + Float32(s * Float32(4.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(x * Float32(x / Float32(s * s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999593223797e-31))
		tmp = single(1.0) / ((x / (s / x)) + (s * single(4.0)));
	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 < 1.99999996e-31

   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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 80.0%

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

   7. Simplified 80.0%

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

      1. clear-num N/A

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

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

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

      3. associate-/r/ N/A

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

      4. /-rgt-identity N/A

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f32 79.9%

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

   9. Applied egg-rr 79.9%

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

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

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

       2. distribute-rgt-in N/A

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

       3. div-inv N/A

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

       4. associate-*l* N/A

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

       5. pow2 N/A

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

       6. pow-flip N/A

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

       7. pow-plus N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. inv-pow N/A

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

       11. div-inv N/A

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

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

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

       13. clear-num N/A

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

       14. associate-/l/ N/A

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

       15. clear-num N/A

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

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

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

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

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

       18. *-commutative N/A

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

       19. *-lowering-*.f32 71.9%

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

   11. Applied egg-rr 71.9%

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

   if 1.99999996e-31 < x

   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. 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. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. distribute-lft-out N/A

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

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

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

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

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(s \cdot \left(4 + \frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)}\right) \]
   10. 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. unpow2 N/A

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

       3. sqr-abs N/A

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

       4. unpow2 N/A

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

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

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

       6. unpow2 N/A

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

       7. associate-/l* N/A

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

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

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f32 86.7%

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

   11. Simplified 86.7%

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

Alternative 9: 77.8% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.239999965066869 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\frac{s \cdot s}{x}}{x}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}} + s \cdot 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= s 1.239999965066869e-23)
   (/ (/ (/ (* s s) x) x) s)
   (/ 1.0 (+ (/ x (/ s x)) (* s 4.0)))))

C

float code(float x, float s) {
	float tmp;
	if (s <= 1.239999965066869e-23f) {
		tmp = (((s * s) / x) / x) / s;
	} else {
		tmp = 1.0f / ((x / (s / x)) + (s * 4.0f));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (s <= 1.239999965066869e-23) then
        tmp = (((s * s) / x) / x) / s
    else
        tmp = 1.0e0 / ((x / (s / x)) + (s * 4.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (s <= Float32(1.239999965066869e-23))
		tmp = Float32(Float32(Float32(Float32(s * s) / x) / x) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x / Float32(s / x)) + Float32(s * Float32(4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (s <= single(1.239999965066869e-23))
		tmp = (((s * s) / x) / x) / s;
	else
		tmp = single(1.0) / ((x / (s / x)) + (s * single(4.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 1.23999997e-23

   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 99.9%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 80.4%

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

   7. Simplified 80.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. associate-/r* N/A

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

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

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

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

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

      7. *-lowering-*.f32 80.4%

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

   9. Applied egg-rr 80.4%

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

   10. Taylor expanded in s around 0

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

       1. unpow2 N/A

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

       2. associate-/r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f32 89.6%

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

   12. Simplified 89.6%

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

   if 1.23999997e-23 < s

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 82.8%

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

   7. Simplified 82.8%

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

      1. clear-num N/A

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

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

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

      3. associate-/r/ N/A

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

      4. /-rgt-identity N/A

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f32 82.8%

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

   9. Applied egg-rr 82.8%

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

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

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

       2. distribute-rgt-in N/A

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

       3. div-inv N/A

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

       4. associate-*l* N/A

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

       5. pow2 N/A

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

       6. pow-flip N/A

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

       7. pow-plus N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. inv-pow N/A

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

       11. div-inv N/A

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

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

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

       13. clear-num N/A

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

       14. associate-/l/ N/A

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

       15. clear-num N/A

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

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

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

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

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

       18. *-commutative N/A

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

       19. *-lowering-*.f32 76.9%

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

   11. Applied egg-rr 76.9%

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

Alternative 10: 50.5% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 5.0000000843119176e-17)
   (/ 0.25 s)
   (/ 1.0 (* s (/ (* x x) (* s s))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 5.0000000843119176e-17f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (s * ((x * x) / (s * s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 5.0000000843119176e-17) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (s * ((x * x) / (s * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(5.0000000843119176e-17))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(x * x) / Float32(s * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(5.0000000843119176e-17))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (s * ((x * x) / (s * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5.00000008e-17

   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 5.00000008e-17 < x

   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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 83.7%

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

   7. Simplified 83.7%

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

      1. clear-num N/A

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

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

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

      3. associate-/r/ N/A

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

      4. /-rgt-identity N/A

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f32 83.7%

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

   9. Applied egg-rr 83.7%

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

   10. Taylor expanded in x around inf

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

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

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

       2. unpow2 N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f32 76.8%

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

   12. Simplified 76.8%

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

Alternative 11: 49.5% accurate, 44.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{s \cdot s}{x}}{x}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 5.0000000843119176e-17) (/ 0.25 s) (/ (/ (/ (* s s) x) x) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 5.0000000843119176e-17f) {
		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 <= 5.0000000843119176e-17) 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(5.0000000843119176e-17))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(Float32(s * s) / x) / x) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(5.0000000843119176e-17))
		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 < 5.00000008e-17

   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 5.00000008e-17 < x

   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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 83.7%

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

   7. Simplified 83.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. associate-/r* N/A

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

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

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

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

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

      7. *-lowering-*.f32 85.8%

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

   9. Applied egg-rr 85.8%

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

   10. Taylor expanded in s around 0

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

       1. unpow2 N/A

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

       2. associate-/r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f32 75.9%

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

   12. Simplified 75.9%

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

Alternative 12: 49.6% accurate, 44.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot s}{x \cdot x}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 5.0000000843119176e-17) (/ 0.25 s) (/ (/ (* s s) (* x x)) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 5.0000000843119176e-17f) {
		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 <= 5.0000000843119176e-17) 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(5.0000000843119176e-17))
		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(5.0000000843119176e-17))
		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 < 5.00000008e-17

   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 5.00000008e-17 < x

   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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 83.7%

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

   7. Simplified 83.7%

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

   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 74.9%

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

   10. Simplified 74.9%

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

Alternative 13: 45.3% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.99999993922529e-9) (/ 0.25 s) (/ 1.0 (/ x (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.99999993922529e-9f) {
		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 <= 9.99999993922529e-9) 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(9.99999993922529e-9))
		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(9.99999993922529e-9))
		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 < 9.99999994e-9

   1. Initial program 99.6%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 39.2%

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

   5. Simplified 39.2%

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

   if 9.99999994e-9 < 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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 86.1%

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

   7. Simplified 86.1%

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

   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 73.9%

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

   10. Simplified 73.9%

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

       1. associate-/r* N/A

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

       2. clear-num N/A

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

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

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

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

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

       5. /-lowering-/.f32 75.1%

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

   12. Applied egg-rr 75.1%

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

Alternative 14: 44.6% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.99999993922529e-9) (/ 0.25 s) (* (/ s x) (/ 1.0 x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.99999993922529e-9f) {
		tmp = 0.25f / s;
	} else {
		tmp = (s / x) * (1.0f / 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 <= 9.99999993922529e-9) then
        tmp = 0.25e0 / s
    else
        tmp = (s / x) * (1.0e0 / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.99999993922529e-9))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(s / x) * Float32(Float32(1.0) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.99999993922529e-9))
		tmp = single(0.25) / s;
	else
		tmp = (s / x) * (single(1.0) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 9.99999994e-9

   1. Initial program 99.6%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 39.2%

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

   5. Simplified 39.2%

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

   if 9.99999994e-9 < 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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 86.1%

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

   7. Simplified 86.1%

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

   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 73.9%

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

   10. Simplified 73.9%

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

       1. associate-/r* N/A

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

       2. div-inv N/A

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

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

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

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

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

       5. /-lowering-/.f32 73.9%

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

   12. Applied egg-rr 73.9%

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

Alternative 15: 44.6% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.99999993922529e-9) (/ 0.25 s) (/ (/ s x) x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.99999993922529e-9f) {
		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 <= 9.99999993922529e-9) 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(9.99999993922529e-9))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(s / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.99999993922529e-9))
		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 < 9.99999994e-9

   1. Initial program 99.6%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 39.2%

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

   5. Simplified 39.2%

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

   if 9.99999994e-9 < 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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 86.1%

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

   7. Simplified 86.1%

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

   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 73.9%

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

   10. Simplified 73.9%

       \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
   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 73.9%

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

   12. Applied egg-rr 73.9%

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

Alternative 16: 44.6% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.99999993922529e-9) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.99999993922529e-9f) {
		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 <= 9.99999993922529e-9) 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(9.99999993922529e-9))
		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(9.99999993922529e-9))
		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 < 9.99999994e-9

   1. Initial program 99.6%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   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 39.2%

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

   5. Simplified 39.2%

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

   if 9.99999994e-9 < 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

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 86.1%

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

   7. Simplified 86.1%

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

   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 73.9%

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

   10. Simplified 73.9%

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

Alternative 17: 27.5% accurate, 206.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. /-lowering-/.f32 29.5%

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

5. Simplified 29.5%

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

Reproduce

herbie shell --seed 2024186 
(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))))))