Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 11.8s

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, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 94.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{x}{-s}}} \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\mathsf{fma}\left(x, \frac{\frac{\mathsf{fma}\left(x, 0.5, 0.16666666666666666 \cdot \frac{x \cdot x}{s}\right)}{s} - -1}{s}, 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ x (- s))))) 2.000000026702864e-10)
   (/
    1.0
    (fma
     x
     (fma (/ x (* s s)) (fma -0.16666666666666666 (/ x s) 0.5) (/ -1.0 s))
     2.0))
   (/
    1.0
    (+
     1.0
     (/
      1.0
      (fma
       x
       (/ (- (/ (fma x 0.5 (* 0.16666666666666666 (/ (* x x) s))) s) -1.0) s)
       1.0))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((x / -s)))) <= 2.000000026702864e-10f) {
		tmp = 1.0f / fmaf(x, fmaf((x / (s * s)), fmaf(-0.16666666666666666f, (x / s), 0.5f), (-1.0f / s)), 2.0f);
	} else {
		tmp = 1.0f / (1.0f + (1.0f / fmaf(x, (((fmaf(x, 0.5f, (0.16666666666666666f * ((x * x) / s))) / s) - -1.0f) / s), 1.0f)));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(x / Float32(-s))))) <= Float32(2.000000026702864e-10))
		tmp = Float32(Float32(1.0) / fma(x, fma(Float32(x / Float32(s * s)), fma(Float32(-0.16666666666666666), Float32(x / s), Float32(0.5)), Float32(Float32(-1.0) / s)), Float32(2.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / fma(x, Float32(Float32(Float32(fma(x, Float32(0.5), Float32(Float32(0.16666666666666666) * Float32(Float32(x * x) / s))) / s) - Float32(-1.0)) / s), Float32(1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 2.00000003e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 91.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}} \]

   if 2.00000003e-10 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   7. Applied rewrites 85.4%

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

   8. Taylor expanded in s around -inf

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

   10. Applied rewrites 95.7%

       \[\leadsto \frac{1}{1 + \frac{1}{\mathsf{fma}\left(x, \frac{-1 - \frac{\mathsf{fma}\left(x, 0.5, \frac{x \cdot x}{s} \cdot 0.16666666666666666\right)}{s}}{\color{blue}{-s}}, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{x}{-s}}} \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\mathsf{fma}\left(x, \frac{\frac{\mathsf{fma}\left(x, 0.5, 0.16666666666666666 \cdot \frac{x \cdot x}{s}\right)}{s} - -1}{s}, 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{x}{-s}}} \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{\mathsf{fma}\left(x, \frac{x}{s} \cdot 0.5, x\right)}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ x (- s))))) 2.000000026702864e-10)
   (/
    1.0
    (fma
     x
     (fma (/ x (* s s)) (fma -0.16666666666666666 (/ x s) 0.5) (/ -1.0 s))
     2.0))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ (fma x (* (/ x s) 0.5) x) s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((x / -s)))) <= 2.000000026702864e-10f) {
		tmp = 1.0f / fmaf(x, fmaf((x / (s * s)), fmaf(-0.16666666666666666f, (x / s), 0.5f), (-1.0f / s)), 2.0f);
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (fmaf(x, ((x / s) * 0.5f), x) / s))));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(x / Float32(-s))))) <= Float32(2.000000026702864e-10))
		tmp = Float32(Float32(1.0) / fma(x, fma(Float32(x / Float32(s * s)), fma(Float32(-0.16666666666666666), Float32(x / s), Float32(0.5)), Float32(Float32(-1.0) / s)), Float32(2.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(fma(x, Float32(Float32(x / s) * Float32(0.5)), x) / s)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 2.00000003e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 91.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}} \]

   if 2.00000003e-10 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in s around -inf

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

      1. lower-+.f32 N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

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

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower-/.f32 N/A

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

   7. Applied rewrites 95.0%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{x}{-s}}} \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{\mathsf{fma}\left(x, \frac{x}{s} \cdot 0.5, x\right)}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{x}{-s}}} \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{2 + \frac{\left(x \cdot -0.16666666666666666\right) \cdot \frac{x \cdot x}{s \cdot s}}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ x (- s))))) 2.000000026702864e-10)
   (/ 1.0 (+ 2.0 (/ (* (* x -0.16666666666666666) (/ (* x x) (* s s))) s)))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((x / -s)))) <= 2.000000026702864e-10f) {
		tmp = 1.0f / (2.0f + (((x * -0.16666666666666666f) * ((x * x) / (s * s))) / s));
	} else {
		tmp = 1.0f / (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 ((1.0e0 / (1.0e0 + exp((x / -s)))) <= 2.000000026702864e-10) then
        tmp = 1.0e0 / (2.0e0 + (((x * (-0.16666666666666666e0)) * ((x * x) / (s * s))) / s))
    else
        tmp = 1.0e0 / (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(1.0) / Float32(Float32(1.0) + exp(Float32(x / Float32(-s))))) <= Float32(2.000000026702864e-10))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(x * Float32(-0.16666666666666666)) * Float32(Float32(x * x) / Float32(s * s))) / s)));
	else
		tmp = Float32(Float32(1.0) / 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 ((single(1.0) / (single(1.0) + exp((x / -s)))) <= single(2.000000026702864e-10))
		tmp = single(1.0) / (single(2.0) + (((x * single(-0.16666666666666666)) * ((x * x) / (s * s))) / s));
	else
		tmp = single(1.0) / (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 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 2.00000003e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in s around -inf

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

      1. lower-+.f32 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 82.0%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\mathsf{fma}\left(\frac{x \cdot \left(x \cdot x\right)}{s}, -0.16666666666666666, 0.5 \cdot \left(x \cdot x\right)\right)}{s} - x}{s}}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{2}}}{s}} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.2%

      \[\leadsto \frac{1}{2 + \frac{\frac{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{s \cdot s}}{s}} \]
   9. Step-by-step derivation

   10. Applied rewrites 86.7%

       \[\leadsto \frac{1}{2 + \frac{\left(x \cdot -0.16666666666666666\right) \cdot \frac{x \cdot x}{s \cdot s}}{s}} \]

   if 2.00000003e-10 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f32 N/A

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

      2. lower-/.f32 92.3

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

   7. Applied rewrites 92.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{x}{-s}}} \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{2 + \frac{\left(x \cdot -0.16666666666666666\right) \cdot \frac{x \cdot x}{s \cdot s}}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{x}{-s}} \leq 2:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \frac{0.5}{s}, 1\right)}{s}, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{\left(x \cdot -0.16666666666666666\right) \cdot \frac{x \cdot x}{s \cdot s}}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ x (- s))) 2.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (fma x (/ (fma x (/ 0.5 s) 1.0) s) 1.0))))
   (/ 1.0 (+ 2.0 (/ (* (* x -0.16666666666666666) (/ (* x x) (* s s))) s)))))

C

float code(float x, float s) {
	float tmp;
	if (expf((x / -s)) <= 2.0f) {
		tmp = 1.0f / (1.0f + (1.0f / fmaf(x, (fmaf(x, (0.5f / s), 1.0f) / s), 1.0f)));
	} else {
		tmp = 1.0f / (2.0f + (((x * -0.16666666666666666f) * ((x * x) / (s * s))) / s));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(x / Float32(-s))) <= Float32(2.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / fma(x, Float32(fma(x, Float32(Float32(0.5) / s), Float32(1.0)) / s), Float32(1.0)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(x * Float32(-0.16666666666666666)) * Float32(Float32(x * x) / Float32(s * s))) / s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   7. Applied rewrites 85.4%

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

   8. Taylor expanded in s around inf

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

   10. Applied rewrites 95.0%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around -inf

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

      1. lower-+.f32 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 82.0%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\mathsf{fma}\left(\frac{x \cdot \left(x \cdot x\right)}{s}, -0.16666666666666666, 0.5 \cdot \left(x \cdot x\right)\right)}{s} - x}{s}}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{2}}}{s}} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.2%

      \[\leadsto \frac{1}{2 + \frac{\frac{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{s \cdot s}}{s}} \]
   9. Step-by-step derivation

   10. Applied rewrites 86.7%

       \[\leadsto \frac{1}{2 + \frac{\left(x \cdot -0.16666666666666666\right) \cdot \frac{x \cdot x}{s \cdot s}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{x}{-s}} \leq 2:\\ \;\;\;\;\frac{1}{1 + \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \frac{0.5}{s}, 1\right)}{s}, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{\left(x \cdot -0.16666666666666666\right) \cdot \frac{x \cdot x}{s \cdot s}}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{x}{-s}}} \leq 0.49799999594688416:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, 0.5, -s\right)}{s \cdot s}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ x (- s))))) 0.49799999594688416)
   (/ 1.0 (fma x (/ (fma x 0.5 (- s)) (* s s)) 2.0))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((x / -s)))) <= 0.49799999594688416f) {
		tmp = 1.0f / fmaf(x, (fmaf(x, 0.5f, -s) / (s * s)), 2.0f);
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(x / Float32(-s))))) <= Float32(0.49799999594688416))
		tmp = Float32(Float32(1.0) / fma(x, Float32(fma(x, Float32(0.5), Float32(-s)) / Float32(s * s)), Float32(2.0)));
	else
		tmp = Float32(Float32(1.0) / 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 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.497999996

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Applied rewrites 75.5%

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 84.5%

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

   if 0.497999996 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 99.9%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f32 N/A

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

      2. lower-/.f32 93.8

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

   7. Applied rewrites 93.8%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{x}{-s}}} \leq 0.49799999594688416:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, 0.5, -s\right)}{s \cdot s}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{x}{-s}}} \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x \cdot 0.5}{s \cdot s}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ x (- s))))) 2.000000026702864e-10)
   (/ 1.0 (fma x (/ (* x 0.5) (* s s)) 2.0))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((x / -s)))) <= 2.000000026702864e-10f) {
		tmp = 1.0f / fmaf(x, ((x * 0.5f) / (s * s)), 2.0f);
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(x / Float32(-s))))) <= Float32(2.000000026702864e-10))
		tmp = Float32(Float32(1.0) / fma(x, Float32(Float32(x * Float32(0.5)) / Float32(s * s)), Float32(2.0)));
	else
		tmp = Float32(Float32(1.0) / 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 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 2.00000003e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 79.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 86.5%

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

   if 2.00000003e-10 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f32 N/A

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

      2. lower-/.f32 92.3

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

   7. Applied rewrites 92.3%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{x}{-s}}} \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x \cdot 0.5}{s \cdot s}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;e^{t\_0} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = x / -s
    if (exp(t_0) <= 2.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (exp(t_0) <= Float32(2.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (exp(t_0) <= single(2.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 51.8%

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

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

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 40.3

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

   5. Applied rewrites 40.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 40.3%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{x}{-s}} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{-s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 10: 91.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.5:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{\mathsf{fma}\left(x, \frac{x}{s} \cdot 0.5, x\right)}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{\left(x \cdot -0.16666666666666666\right) \cdot \frac{x \cdot x}{s \cdot s}}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 0.5)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ (fma x (* (/ x s) 0.5) x) s)))))
   (/ 1.0 (+ 2.0 (/ (* (* x -0.16666666666666666) (/ (* x x) (* s s))) s)))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 0.5f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (fmaf(x, ((x / s) * 0.5f), x) / s))));
	} else {
		tmp = 1.0f / (2.0f + (((x * -0.16666666666666666f) * ((x * x) / (s * s))) / s));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(0.5))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(fma(x, Float32(Float32(x / s) * Float32(0.5)), x) / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(x * Float32(-0.16666666666666666)) * Float32(Float32(x * x) / Float32(s * s))) / s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in s around -inf

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

      1. lower-+.f32 N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

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

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower-/.f32 N/A

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

   7. Applied rewrites 95.0%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around -inf

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

      1. lower-+.f32 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 82.0%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\mathsf{fma}\left(\frac{x \cdot \left(x \cdot x\right)}{s}, -0.16666666666666666, 0.5 \cdot \left(x \cdot x\right)\right)}{s} - x}{s}}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2 + \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{2}}}{s}} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.2%

      \[\leadsto \frac{1}{2 + \frac{\frac{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{s \cdot s}}{s}} \]
   9. Step-by-step derivation

   10. Applied rewrites 86.7%

       \[\leadsto \frac{1}{2 + \frac{\left(x \cdot -0.16666666666666666\right) \cdot \frac{x \cdot x}{s \cdot s}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.5:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{\mathsf{fma}\left(x, \frac{x}{s} \cdot 0.5, x\right)}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{\left(x \cdot -0.16666666666666666\right) \cdot \frac{x \cdot x}{s \cdot s}}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 51.8%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Applied rewrites 75.9%

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

   7. Applied rewrites 79.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 86.5%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x \cdot 0.5}{s \cdot s}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 51.8%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 83.3%

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

4. Final simplification 63.2%

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

Alternative 13: 49.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.800000011920929:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{-1}{s}, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -0.800000011920929) 0.5 (/ 1.0 (fma x (/ -1.0 s) 2.0))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -0.800000011920929f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, (-1.0f / s), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(-0.800000011920929))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(Float32(-1.0) / s), Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -0.800000012

   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 28.2%

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

   if -0.800000012 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Applied rewrites 83.8%

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

   7. Applied rewrites 86.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.4%

       \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{-1}{s}, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.800000011920929:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{-1}{s}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.800000011920929:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -0.800000011920929) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -0.800000011920929f) {
		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) <= (-0.800000011920929e0)) 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(x / Float32(-s)) <= Float32(-0.800000011920929))
		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(-0.800000011920929))
		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) < -0.800000012

   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 28.2%

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

   if -0.800000012 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 61.8

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

   5. Applied rewrites 61.8%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.800000011920929:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.3% 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 99.9%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
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 2024222 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))