Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 7.1s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{\frac{-x}{s}}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + expf((-x / s)));
}

Fortran

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))
end

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{\frac{-x}{s}}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + expf((-x / s)));
}

Fortran

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))
end

MATLAB

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{e^{\frac{x}{s}}}}\\ \frac{1}{1 + t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (sqrt (/ 1.0 (exp (/ x s)))))) (/ 1.0 (+ 1.0 (* t_0 t_0)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. sqrt-pow1 N/A

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

   4. pow2 N/A

      \[\leadsto \frac{1}{1 + \sqrt{\color{blue}{e^{\frac{-x}{s}} \cdot e^{\frac{-x}{s}}}}} \]

   5. sqr-neg-rev N/A

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

   6. sqr-neg-rev N/A

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

   7. sqrt-prod N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 99.7%

   \[\leadsto \frac{1}{1 + \color{blue}{\sqrt{-\frac{-1}{e^{\frac{x}{s}}}} \cdot \sqrt{-\frac{-1}{e^{\frac{x}{s}}}}}} \]

5. Final simplification 99.7%

   \[\leadsto \frac{1}{1 + \sqrt{\frac{1}{e^{\frac{x}{s}}}} \cdot \sqrt{\frac{1}{e^{\frac{x}{s}}}}} \]
6. Add Preprocessing

Alternative 2: 89.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.0000000200408773 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{1 - \frac{-1}{\frac{x}{s} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{-1}{s} + \frac{2}{x}}{x} + \frac{0.5}{s \cdot s}\right) \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 1.0000000200408773e+20)
   (/ 1.0 (- 1.0 (/ -1.0 (+ (/ x s) 1.0))))
   (/ 1.0 (* (+ (/ (+ (/ -1.0 s) (/ 2.0 x)) x) (/ 0.5 (* s s))) (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.0000000200408773e+20f) {
		tmp = 1.0f / (1.0f - (-1.0f / ((x / s) + 1.0f)));
	} else {
		tmp = 1.0f / (((((-1.0f / s) + (2.0f / x)) / x) + (0.5f / (s * 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 ((1.0e0 + exp((-x / s))) <= 1.0000000200408773e+20) then
        tmp = 1.0e0 / (1.0e0 - ((-1.0e0) / ((x / s) + 1.0e0)))
    else
        tmp = 1.0e0 / ((((((-1.0e0) / s) + (2.0e0 / x)) / x) + (0.5e0 / (s * s))) * (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(1.0000000200408773e+20))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(-1.0) / Float32(Float32(x / s) + Float32(1.0)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(Float32(-1.0) / s) + Float32(Float32(2.0) / x)) / x) + Float32(Float32(0.5) / Float32(s * s))) * Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) + exp((-x / s))) <= single(1.0000000200408773e+20))
		tmp = single(1.0) / (single(1.0) - (single(-1.0) / ((x / s) + single(1.0))));
	else
		tmp = single(1.0) / (((((single(-1.0) / s) + (single(2.0) / x)) / x) + (single(0.5) / (s * s))) * (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.00000002e20

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{1 + e^{\frac{-x}{s}}}} \]

      2. *-lft-identity N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

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

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

      5. *-lft-identity N/A

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

      6. lower--.f32 N/A

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

      7. lift-exp.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-neg.f32 N/A

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

      10. distribute-frac-neg N/A

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

      11. exp-neg N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lift-neg.f32 N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-/.f32 N/A

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

      18. lower-/.f32 N/A

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

      19. lift-/.f32 N/A

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

      20. distribute-frac-neg N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 94.3

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

   7. Applied rewrites 94.3%

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

   if 1.00000002e20 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 6.3

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \color{blue}{\frac{1}{s}}, x, 2\right)} \]

   5. Applied rewrites 6.4%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \frac{1}{s}, x, 2\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{-1}{s}, x, 2\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 6.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{-1}{s}, x, 2\right)} \]

   9. Taylor expanded in x around -inf

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

   11. Applied rewrites 89.4%

       \[\leadsto \frac{1}{\left(\frac{\frac{1}{s} - \frac{2}{x}}{-x} + \frac{0.5}{s \cdot s}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.0000000200408773 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{1 - \frac{-1}{\frac{x}{s} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{-1}{s} + \frac{2}{x}}{x} + \frac{0.5}{s \cdot s}\right) \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.0000000200408773 \cdot 10^{+20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.5}{s \cdot s} \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 1.0000000200408773e+20)
   0.5
   (/ 1.0 (* (/ 0.5 (* s s)) (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.0000000200408773e+20f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((0.5f / (s * 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 ((1.0e0 + exp((-x / s))) <= 1.0000000200408773e+20) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((0.5e0 / (s * s)) * (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(1.0000000200408773e+20))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(0.5) / Float32(s * s)) * Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) + exp((-x / s))) <= single(1.0000000200408773e+20))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((single(0.5) / (s * s)) * (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.00000002e20

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 53.8%

      \[\leadsto \color{blue}{0.5} \]

   if 1.00000002e20 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 6.3

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \color{blue}{\frac{1}{s}}, x, 2\right)} \]

   5. Applied rewrites 6.4%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \frac{1}{s}, x, 2\right)}} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 89.4%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\frac{\frac{1}{2}}{{s}^{2}} \cdot \left(x \cdot x\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.4%

       \[\leadsto \frac{1}{\frac{0.5}{s \cdot s} \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.0000000200408773 \cdot 10^{+20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.5}{s \cdot s} \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.0000000200408773 \cdot 10^{+20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(-x\right) \cdot s}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 1.0000000200408773e+20)
   0.5
   (/ 1.0 (/ (* (- x) s) (* s s)))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.0000000200408773e+20f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((-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 ((1.0e0 + exp((-x / s))) <= 1.0000000200408773e+20) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((-x * s) / (s * s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(1.0000000200408773e+20))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(-x) * s) / Float32(s * s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) + exp((-x / s))) <= single(1.0000000200408773e+20))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((-x * s) / (s * s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.00000002e20

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 53.8%

      \[\leadsto \color{blue}{0.5} \]

   if 1.00000002e20 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 6.3

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \color{blue}{\frac{1}{s}}, x, 2\right)} \]

   5. Applied rewrites 6.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \frac{1}{s}, x, 2\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.0%

      \[\leadsto \frac{1}{\frac{\frac{0.5 \cdot x}{s} - 1}{s} \cdot x + \color{blue}{2}} \]

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 85.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 50.7%

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

Alternative 5: 48.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.5:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(\frac{-1}{s}, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1}{s} \cdot x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 1.5)
   (/ 1.0 (+ 1.0 (fma (/ -1.0 s) x 1.0)))
   (/ 1.0 (+ (* (/ -1.0 s) x) 2.0))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.5f) {
		tmp = 1.0f / (1.0f + fmaf((-1.0f / s), x, 1.0f));
	} else {
		tmp = 1.0f / (((-1.0f / s) * x) + 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(1.5))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + fma(Float32(Float32(-1.0) / s), x, Float32(1.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(-1.0) / s) * x) + Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.5

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{1 + \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{1 + \left(\color{blue}{\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) \cdot x} + 1\right)} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, x, 1\right)}} \]

   5. Applied rewrites 28.9%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right)}{s \cdot s} \cdot x - \frac{1}{s}, x, 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{-1}{s}, x, 1\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 27.9%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{-1}{s}, x, 1\right)} \]

   if 1.5 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.5%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 44.3

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \color{blue}{\frac{1}{s}}, x, 2\right)} \]

   5. Applied rewrites 43.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \frac{1}{s}, x, 2\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{-1}{s}, x, 2\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.3%

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

   10. Applied rewrites 63.8%

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

Alternative 6: 48.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1}{s} \cdot x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 1.5) 0.5 (/ 1.0 (+ (* (/ -1.0 s) x) 2.0))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.5f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (((-1.0f / s) * x) + 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((1.0e0 + exp((-x / s))) <= 1.5e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((((-1.0e0) / s) * x) + 2.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(1.5))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(-1.0) / s) * x) + Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) + exp((-x / s))) <= single(1.5))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (((single(-1.0) / s) * x) + single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.5

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 28.1%

      \[\leadsto \color{blue}{0.5} \]

   if 1.5 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.5%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 44.3

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \color{blue}{\frac{1}{s}}, x, 2\right)} \]

   5. Applied rewrites 43.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \frac{1}{s}, x, 2\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{-1}{s}, x, 2\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.8%

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

   10. Applied rewrites 63.8%

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

Alternative 7: 48.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 1.5) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.5f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f - (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 ((1.0e0 + exp((-x / s))) <= 1.5e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(1.5))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) + exp((-x / s))) <= single(1.5))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.5

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 28.1%

      \[\leadsto \color{blue}{0.5} \]

   if 1.5 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.5%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto \frac{1}{2 - \color{blue}{1} \cdot \frac{x}{s}} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

      4. lower--.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

      5. lower-/.f32 63.8

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x}{s}}} \]

   5. Applied rewrites 63.8%

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

Alternative 8: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 - \frac{-1}{e^{\frac{x}{s}}}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (- 1.0 (/ -1.0 (exp (/ x s))))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{1 + e^{\frac{-x}{s}}}} \]

   2. *-lft-identity N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

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

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

   5. *-lft-identity N/A

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

   6. lower--.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. distribute-frac-neg N/A

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

   11. exp-neg N/A

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

   12. distribute-neg-frac N/A

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

   13. metadata-eval N/A

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

   14. remove-double-neg N/A

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

   15. lift-neg.f32 N/A

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

   16. distribute-frac-neg N/A

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

   17. lift-/.f32 N/A

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

   18. lower-/.f32 N/A

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

   19. lift-/.f32 N/A

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

   20. distribute-frac-neg N/A

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

4. Applied rewrites 99.7%

   \[\leadsto \frac{1}{\color{blue}{1 - \frac{-1}{e^{\frac{x}{s}}}}} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{\frac{-x}{s}}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + expf((-x / s)));
}

Fortran

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))
end

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 10: 89.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 50:\\ \;\;\;\;\frac{1}{1 - \frac{-1}{\frac{x}{s} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s}}{x}\right) \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 50.0)
   (/ 1.0 (- 1.0 (/ -1.0 (+ (/ x s) 1.0))))
   (/ 1.0 (* (- (/ 0.5 (* s s)) (/ (/ 1.0 s) x)) (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 50.0f) {
		tmp = 1.0f / (1.0f - (-1.0f / ((x / s) + 1.0f)));
	} else {
		tmp = 1.0f / (((0.5f / (s * s)) - ((1.0f / s) / x)) * (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 50.0e0) then
        tmp = 1.0e0 / (1.0e0 - ((-1.0e0) / ((x / s) + 1.0e0)))
    else
        tmp = 1.0e0 / (((0.5e0 / (s * s)) - ((1.0e0 / s) / x)) * (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(50.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(-1.0) / Float32(Float32(x / s) + Float32(1.0)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) - Float32(Float32(Float32(1.0) / s) / x)) * Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(50.0))
		tmp = single(1.0) / (single(1.0) - (single(-1.0) / ((x / s) + single(1.0))));
	else
		tmp = single(1.0) / (((single(0.5) / (s * s)) - ((single(1.0) / s) / x)) * (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{1 + e^{\frac{-x}{s}}}} \]

      2. *-lft-identity N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

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

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

      5. *-lft-identity N/A

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

      6. lower--.f32 N/A

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

      7. lift-exp.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-neg.f32 N/A

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

      10. distribute-frac-neg N/A

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

      11. exp-neg N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lift-neg.f32 N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-/.f32 N/A

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

      18. lower-/.f32 N/A

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

      19. lift-/.f32 N/A

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

      20. distribute-frac-neg N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 94.3

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

   7. Applied rewrites 94.3%

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

   if 50 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 6.3

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \color{blue}{\frac{1}{s}}, x, 2\right)} \]

   5. Applied rewrites 6.4%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \frac{1}{s}, x, 2\right)}} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 89.4%

      \[\leadsto \frac{1}{\left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s}}{x}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 50:\\ \;\;\;\;\frac{1}{1 - \frac{-1}{\frac{x}{s} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s}}{x}\right) \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 50:\\ \;\;\;\;\frac{1}{1 - \frac{-1}{\frac{x}{s} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.5}{s \cdot s} \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 50.0)
   (/ 1.0 (- 1.0 (/ -1.0 (+ (/ x s) 1.0))))
   (/ 1.0 (* (/ 0.5 (* s s)) (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 50.0f) {
		tmp = 1.0f / (1.0f - (-1.0f / ((x / s) + 1.0f)));
	} else {
		tmp = 1.0f / ((0.5f / (s * 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 / s) <= 50.0e0) then
        tmp = 1.0e0 / (1.0e0 - ((-1.0e0) / ((x / s) + 1.0e0)))
    else
        tmp = 1.0e0 / ((0.5e0 / (s * s)) * (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(50.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(-1.0) / Float32(Float32(x / s) + Float32(1.0)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(0.5) / Float32(s * s)) * Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(50.0))
		tmp = single(1.0) / (single(1.0) - (single(-1.0) / ((x / s) + single(1.0))));
	else
		tmp = single(1.0) / ((single(0.5) / (s * s)) * (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{1 + e^{\frac{-x}{s}}}} \]

      2. *-lft-identity N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

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

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

      5. *-lft-identity N/A

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

      6. lower--.f32 N/A

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

      7. lift-exp.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. lift-neg.f32 N/A

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

      10. distribute-frac-neg N/A

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

      11. exp-neg N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lift-neg.f32 N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-/.f32 N/A

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

      18. lower-/.f32 N/A

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

      19. lift-/.f32 N/A

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

      20. distribute-frac-neg N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 94.3

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

   7. Applied rewrites 94.3%

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

   if 50 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 6.3

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \color{blue}{\frac{1}{s}}, x, 2\right)} \]

   5. Applied rewrites 6.4%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \frac{1}{s}, x, 2\right)}} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 89.4%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\frac{\frac{1}{2}}{{s}^{2}} \cdot \left(x \cdot x\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.4%

       \[\leadsto \frac{1}{\frac{0.5}{s \cdot s} \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 50:\\ \;\;\;\;\frac{1}{1 - \frac{-1}{\frac{x}{s} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.5}{s \cdot s} \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s))) (if (<= t_0 0.5) 0.5 (/ 1.0 t_0))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 0.5f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= 0.5e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.5))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(0.5))
		tmp = single(0.5);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.5

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 54.5%

      \[\leadsto \color{blue}{0.5} \]

   if 0.5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 6.5

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \color{blue}{\frac{1}{s}}, x, 2\right)} \]

   5. Applied rewrites 6.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{0.5}{s \cdot s} \cdot x - \frac{1}{s}, x, 2\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.5%

      \[\leadsto \frac{1}{\frac{\frac{0.5 \cdot x}{s} - 1}{s} \cdot x + \color{blue}{2}} \]

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 82.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 37.6%

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

Alternative 13: 34.9% accurate, 128.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 0.5)

C

float code(float x, float s) {
	return 0.5f;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0
end function

Julia

function code(x, s)
	return Float32(0.5)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5);
end

Derivation

1. Initial program 99.7%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 38.2%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))