Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 9.4s

Alternatives: 15

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, 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 2: 91.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{-x}{s}}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{1 + \left(\left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s} - \frac{1}{x}}{x}\right) \cdot x\right) \cdot x}\\ \mathbf{elif}\;t\_0 \leq 0.800000011920929:\\ \;\;\;\;\frac{1}{1 + \left(\left(1 + \left(\frac{x}{s} \cdot x\right) \cdot \frac{0.5}{s}\right) - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ (- x) s))))))
   (if (<= t_0 0.0)
     (/
      1.0
      (+ 1.0 (* (* (- (/ 0.5 (* s s)) (/ (- (/ 1.0 s) (/ 1.0 x)) x)) x) x)))
     (if (<= t_0 0.800000011920929)
       (/ 1.0 (+ 1.0 (- (+ 1.0 (* (* (/ x s) x) (/ 0.5 s))) (/ x s))))
       (/ 1.0 (fma (- 1.0 (/ x s)) 1.0 1.0))))))

C

float code(float x, float s) {
	float t_0 = 1.0f / (1.0f + expf((-x / s)));
	float tmp;
	if (t_0 <= 0.0f) {
		tmp = 1.0f / (1.0f + ((((0.5f / (s * s)) - (((1.0f / s) - (1.0f / x)) / x)) * x) * x));
	} else if (t_0 <= 0.800000011920929f) {
		tmp = 1.0f / (1.0f + ((1.0f + (((x / s) * x) * (0.5f / s))) - (x / s)));
	} else {
		tmp = 1.0f / fmaf((1.0f - (x / s)), 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) - Float32(Float32(Float32(Float32(1.0) / s) - Float32(Float32(1.0) / x)) / x)) * x) * x)));
	elseif (t_0 <= Float32(0.800000011920929))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(Float32(Float32(x / s) * x) * Float32(Float32(0.5) / s))) - Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(1.0) - Float32(x / s)), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.0

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 78.2%

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

   if 0.0 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.800000012

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 95.3%

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

   if 0.800000012 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.2

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

   5. Applied rewrites 5.2%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.3

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

   7. Applied rewrites 98.1%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0:\\ \;\;\;\;\frac{1}{1 + \left(\left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s} - \frac{1}{x}}{x}\right) \cdot x\right) \cdot x}\\ \mathbf{elif}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.800000011920929:\\ \;\;\;\;\frac{1}{1 + \left(\left(1 + \left(\frac{x}{s} \cdot x\right) \cdot \frac{0.5}{s}\right) - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{-x}{s}}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{1 + \left(0.5 \cdot x - s\right) \cdot \frac{x}{s \cdot s}}\\ \mathbf{elif}\;t\_0 \leq 0.800000011920929:\\ \;\;\;\;\frac{1}{1 + \left(\left(1 + \left(\frac{x}{s} \cdot x\right) \cdot \frac{0.5}{s}\right) - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ (- x) s))))))
   (if (<= t_0 0.0)
     (/ 1.0 (+ 1.0 (* (- (* 0.5 x) s) (/ x (* s s)))))
     (if (<= t_0 0.800000011920929)
       (/ 1.0 (+ 1.0 (- (+ 1.0 (* (* (/ x s) x) (/ 0.5 s))) (/ x s))))
       (/ 1.0 (fma (- 1.0 (/ x s)) 1.0 1.0))))))

C

float code(float x, float s) {
	float t_0 = 1.0f / (1.0f + expf((-x / s)));
	float tmp;
	if (t_0 <= 0.0f) {
		tmp = 1.0f / (1.0f + (((0.5f * x) - s) * (x / (s * s))));
	} else if (t_0 <= 0.800000011920929f) {
		tmp = 1.0f / (1.0f + ((1.0f + (((x / s) * x) * (0.5f / s))) - (x / s)));
	} else {
		tmp = 1.0f / fmaf((1.0f - (x / s)), 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(Float32(0.5) * x) - s) * Float32(x / Float32(s * s)))));
	elseif (t_0 <= Float32(0.800000011920929))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(Float32(Float32(x / s) * x) * Float32(Float32(0.5) / s))) - Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(1.0) - Float32(x / s)), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.0

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 70.2%

      \[\leadsto \frac{1}{1 + \frac{\left(x \cdot x\right) \cdot 0.5 - s \cdot x}{\color{blue}{s \cdot s}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 70.2%

       \[\leadsto \frac{1}{1 + \frac{x \cdot \left(0.5 \cdot x - s\right)}{s \cdot s}} \]
   11. Step-by-step derivation

   12. Applied rewrites 76.5%

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

   if 0.0 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.800000012

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 95.3%

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

   if 0.800000012 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.2

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

   5. Applied rewrites 5.2%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.3

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

   7. Applied rewrites 98.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0:\\ \;\;\;\;\frac{1}{1 + \left(0.5 \cdot x - s\right) \cdot \frac{x}{s \cdot s}}\\ \mathbf{elif}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.800000011920929:\\ \;\;\;\;\frac{1}{1 + \left(\left(1 + \left(\frac{x}{s} \cdot x\right) \cdot \frac{0.5}{s}\right) - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ t_1 := 1 - \frac{x}{s}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(t\_1, 1, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;\frac{1}{1 + \left(\left(\frac{x}{s} \cdot x\right) \cdot \frac{0.5}{s} + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(0.5 \cdot x - s\right) \cdot \frac{x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)) (t_1 (- 1.0 (/ x s))))
   (if (<= t_0 -2.0)
     (/ 1.0 (fma t_1 1.0 1.0))
     (if (<= t_0 20.0)
       (/ 1.0 (+ 1.0 (+ (* (* (/ x s) x) (/ 0.5 s)) t_1)))
       (/ 1.0 (+ 1.0 (* (- (* 0.5 x) s) (/ x (* s s)))))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float t_1 = 1.0f - (x / s);
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f / fmaf(t_1, 1.0f, 1.0f);
	} else if (t_0 <= 20.0f) {
		tmp = 1.0f / (1.0f + ((((x / s) * x) * (0.5f / s)) + t_1));
	} else {
		tmp = 1.0f / (1.0f + (((0.5f * x) - s) * (x / (s * s))));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	t_1 = Float32(Float32(1.0) - Float32(x / s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = Float32(Float32(1.0) / fma(t_1, Float32(1.0), Float32(1.0)));
	elseif (t_0 <= Float32(20.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(Float32(x / s) * x) * Float32(Float32(0.5) / s)) + t_1)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(Float32(0.5) * x) - s) * Float32(x / Float32(s * s)))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.2

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

   5. Applied rewrites 5.2%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.3

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

   7. Applied rewrites 98.1%

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

   if -2 < (/.f32 (neg.f32 x) s) < 20

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 95.2%

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

   if 20 < (/.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

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 70.2%

      \[\leadsto \frac{1}{1 + \frac{\left(x \cdot x\right) \cdot 0.5 - s \cdot x}{\color{blue}{s \cdot s}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 70.2%

       \[\leadsto \frac{1}{1 + \frac{x \cdot \left(0.5 \cdot x - s\right)}{s \cdot s}} \]
   11. Step-by-step derivation

   12. Applied rewrites 76.5%

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

Alternative 5: 91.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;\frac{1}{1 + \left(1 - \frac{x - \left(0.5 \cdot \frac{x}{s}\right) \cdot x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(0.5 \cdot x - s\right) \cdot \frac{x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0)
     (/ 1.0 (fma (- 1.0 (/ x s)) 1.0 1.0))
     (if (<= t_0 20.0)
       (/ 1.0 (+ 1.0 (- 1.0 (/ (- x (* (* 0.5 (/ x s)) x)) s))))
       (/ 1.0 (+ 1.0 (* (- (* 0.5 x) s) (/ x (* s s)))))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f / fmaf((1.0f - (x / s)), 1.0f, 1.0f);
	} else if (t_0 <= 20.0f) {
		tmp = 1.0f / (1.0f + (1.0f - ((x - ((0.5f * (x / s)) * x)) / s)));
	} else {
		tmp = 1.0f / (1.0f + (((0.5f * x) - s) * (x / (s * s))));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(1.0) - Float32(x / s)), Float32(1.0), Float32(1.0)));
	elseif (t_0 <= Float32(20.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) - Float32(Float32(x - Float32(Float32(Float32(0.5) * Float32(x / s)) * x)) / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(Float32(0.5) * x) - s) * Float32(x / Float32(s * s)))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.2

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

   5. Applied rewrites 5.2%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.3

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

   7. Applied rewrites 98.1%

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

   if -2 < (/.f32 (neg.f32 x) s) < 20

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 84.6%

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

   7. Applied rewrites 95.2%

      \[\leadsto \frac{1}{1 + \left(\left(\frac{x}{s} \cdot x\right) \cdot \frac{0.5}{s} + \color{blue}{\left(1 - \frac{x}{s}\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 95.1%

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

   if 20 < (/.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

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 70.2%

      \[\leadsto \frac{1}{1 + \frac{\left(x \cdot x\right) \cdot 0.5 - s \cdot x}{\color{blue}{s \cdot s}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 70.2%

       \[\leadsto \frac{1}{1 + \frac{x \cdot \left(0.5 \cdot x - s\right)}{s \cdot s}} \]
   11. Step-by-step derivation

   12. Applied rewrites 76.5%

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

Alternative 6: 90.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ t_1 := 1 - \frac{x}{s}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(t\_1, 1, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;\frac{1}{1 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(0.5 \cdot x - s\right) \cdot \frac{x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)) (t_1 (- 1.0 (/ x s))))
   (if (<= t_0 -2.0)
     (/ 1.0 (fma t_1 1.0 1.0))
     (if (<= t_0 20.0)
       (/ 1.0 (+ 1.0 t_1))
       (/ 1.0 (+ 1.0 (* (- (* 0.5 x) s) (/ x (* s s)))))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float t_1 = 1.0f - (x / s);
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f / fmaf(t_1, 1.0f, 1.0f);
	} else if (t_0 <= 20.0f) {
		tmp = 1.0f / (1.0f + t_1);
	} else {
		tmp = 1.0f / (1.0f + (((0.5f * x) - s) * (x / (s * s))));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	t_1 = Float32(Float32(1.0) - Float32(x / s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = Float32(Float32(1.0) / fma(t_1, Float32(1.0), Float32(1.0)));
	elseif (t_0 <= Float32(20.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + t_1));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(Float32(0.5) * x) - s) * Float32(x / Float32(s * s)))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.2

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

   5. Applied rewrites 5.2%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.3

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

   7. Applied rewrites 98.1%

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

   if -2 < (/.f32 (neg.f32 x) s) < 20

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 92.7

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

   5. Applied rewrites 92.7%

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

   if 20 < (/.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

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 70.2%

      \[\leadsto \frac{1}{1 + \frac{\left(x \cdot x\right) \cdot 0.5 - s \cdot x}{\color{blue}{s \cdot s}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 70.2%

       \[\leadsto \frac{1}{1 + \frac{x \cdot \left(0.5 \cdot x - s\right)}{s \cdot s}} \]
   11. Step-by-step derivation

   12. Applied rewrites 76.5%

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

Alternative 7: 88.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ t_1 := 1 - \frac{x}{s}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(t\_1, 1, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 10000000000:\\ \;\;\;\;\frac{1}{1 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{\left(0.5 \cdot x\right) \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)) (t_1 (- 1.0 (/ x s))))
   (if (<= t_0 -2.0)
     (/ 1.0 (fma t_1 1.0 1.0))
     (if (<= t_0 10000000000.0)
       (/ 1.0 (+ 1.0 t_1))
       (/ 1.0 (+ 1.0 (/ (* (* 0.5 x) x) (* s s))))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float t_1 = 1.0f - (x / s);
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f / fmaf(t_1, 1.0f, 1.0f);
	} else if (t_0 <= 10000000000.0f) {
		tmp = 1.0f / (1.0f + t_1);
	} else {
		tmp = 1.0f / (1.0f + (((0.5f * x) * x) / (s * s)));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	t_1 = Float32(Float32(1.0) - Float32(x / s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = Float32(Float32(1.0) / fma(t_1, Float32(1.0), Float32(1.0)));
	elseif (t_0 <= Float32(10000000000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + t_1));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(Float32(0.5) * x) * x) / Float32(s * s))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.2

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

   5. Applied rewrites 5.2%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.3

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

   7. Applied rewrites 98.1%

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

   if -2 < (/.f32 (neg.f32 x) s) < 1e10

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 77.8

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

   5. Applied rewrites 77.8%

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

   if 1e10 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 81.8%

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

Alternative 8: 79.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ t_1 := 1 - \frac{x}{s}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(t\_1, 1, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 20000000000:\\ \;\;\;\;\frac{1}{1 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{\left(-s\right) \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)) (t_1 (- 1.0 (/ x s))))
   (if (<= t_0 -2.0)
     (/ 1.0 (fma t_1 1.0 1.0))
     (if (<= t_0 20000000000.0)
       (/ 1.0 (+ 1.0 t_1))
       (/ 1.0 (+ 1.0 (/ (* (- s) x) (* s s))))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float t_1 = 1.0f - (x / s);
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f / fmaf(t_1, 1.0f, 1.0f);
	} else if (t_0 <= 20000000000.0f) {
		tmp = 1.0f / (1.0f + t_1);
	} else {
		tmp = 1.0f / (1.0f + ((-s * x) / (s * s)));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	t_1 = Float32(Float32(1.0) - Float32(x / s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = Float32(Float32(1.0) / fma(t_1, Float32(1.0), Float32(1.0)));
	elseif (t_0 <= Float32(20000000000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + t_1));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(-s) * x) / Float32(s * s))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.2

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

   5. Applied rewrites 5.2%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.3

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

   7. Applied rewrites 98.1%

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

   if -2 < (/.f32 (neg.f32 x) s) < 2e10

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 77.0

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

   5. Applied rewrites 77.0%

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

   if 2e10 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 82.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 58.3%

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

Alternative 9: 75.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{s}\\ \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(t\_0, 1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (- 1.0 (/ x s))))
   (if (<= (/ (- x) s) -2.0) (/ 1.0 (fma t_0 1.0 1.0)) (/ 1.0 (+ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = 1.0f - (x / s);
	float tmp;
	if ((-x / s) <= -2.0f) {
		tmp = 1.0f / fmaf(t_0, 1.0f, 1.0f);
	} else {
		tmp = 1.0f / (1.0f + t_0);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) - Float32(x / s))
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-2.0))
		tmp = Float32(Float32(1.0) / fma(t_0, Float32(1.0), Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.2

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

   5. Applied rewrites 5.2%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.3

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

   7. Applied rewrites 98.1%

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

   if -2 < (/.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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 62.3

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

   5. Applied rewrites 62.3%

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

Alternative 10: 75.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{s}\\ \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, t\_0, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (- 1.0 (/ x s))))
   (if (<= (/ (- x) s) -2.0) (/ 1.0 (fma 1.0 t_0 1.0)) (/ 1.0 (+ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = 1.0f - (x / s);
	float tmp;
	if ((-x / s) <= -2.0f) {
		tmp = 1.0f / fmaf(1.0f, t_0, 1.0f);
	} else {
		tmp = 1.0f / (1.0f + t_0);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) - Float32(x / s))
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-2.0))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), t_0, Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.2

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

   5. Applied rewrites 5.2%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.3

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

   7. Applied rewrites 98.2%

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

   if -2 < (/.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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 62.3

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

   5. Applied rewrites 62.3%

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

Alternative 11: 47.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.2

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

   5. Applied rewrites 5.2%

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

   7. Applied rewrites 28.1%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 29.0%

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

   if -2 < (/.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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 62.3

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

   5. Applied rewrites 62.3%

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

Alternative 12: 46.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.2

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

   5. Applied rewrites 5.2%

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

   7. Applied rewrites 29.0%

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

   if -2 < (/.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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 62.3

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

   5. Applied rewrites 62.3%

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

Alternative 13: 49.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   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 -2 < (/.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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 62.3

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

   5. Applied rewrites 62.3%

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

Alternative 14: 49.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -2.0f) {
		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 ((-x / s) <= (-2.0e0)) 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(-x) / s) <= Float32(-2.0))
		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 ((-x / s) <= single(-2.0))
		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 (neg.f32 x) s) < -2

   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 -2 < (/.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. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 62.2

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

   5. Applied rewrites 62.2%

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

Alternative 15: 35.2% 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 35.3%

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

Reproduce

herbie shell --seed 2024318 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))