Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 10.0s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 66.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 0.029999999329447746:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 0.029999999329447746)
   0.5
   (/
    1.0
    (fma
     x
     (fma (/ x (* s s)) (fma -0.16666666666666666 (/ x s) 0.5) (/ -1.0 s))
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 0.029999999329447746f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, fmaf((x / (s * s)), fmaf(-0.16666666666666666f, (x / s), 0.5f), (-1.0f / s)), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(0.029999999329447746))
		tmp = Float32(0.5);
	else
		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)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.0299999993

   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 51.5%

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

   if 0.0299999993 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   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 90.1%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 0.029999999329447746:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) -5.0)
   0.5
   (/ 1.0 (fma x (/ (fma (/ x s) 0.5 -1.0) s) 2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= -5.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, (fmaf((x / s), 0.5f, -1.0f) / s), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(-5.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(fma(Float32(x / s), Float32(0.5), 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) < -5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

   if -5 < (/.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 85.1%

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

      1. lift-/.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift-/.f32 85.1

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

      6. lift-fma.f32 N/A

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

      7. lift-/.f32 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-/.f32 89.0

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

      12. lift-fma.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 89.0

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

   7. Applied rewrites 89.0%

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

4. Final simplification 65.7%

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

Alternative 4: 62.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(0.5))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(fma(Float32(0.5), Float32(x / s), Float32(-1.0)) / 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. Taylor expanded in x around 0

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

   5. Applied rewrites 51.3%

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

   if 0.5 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)}} \]
   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 77.6%

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

      1. lift-/.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift-/.f32 77.6

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

      6. lift-fma.f32 N/A

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

      7. lift-/.f32 N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-/.f32 84.1

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

      12. lift-fma.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 84.1

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-in N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. associate-/l/ N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-/r* N/A

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

      16. associate-*r/ N/A

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

   10. Applied rewrites 84.1%

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

4. Final simplification 63.2%

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

Alternative 5: 63.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot s}{x} \cdot \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 20.0) 0.5 (* (/ (* s s) x) (/ 2.0 x))))

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 20.0e0) then
        tmp = 0.5e0
    else
        tmp = ((s * s) / x) * (2.0e0 / x)
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(20.0))
		tmp = single(0.5);
	else
		tmp = ((s * s) / x) * (single(2.0) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   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 50.9%

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

   if 20 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)}} \]
   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 78.8%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 82.1

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

   8. Applied rewrites 82.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. /-rgt-identity N/A

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

      6. /-rgt-identity N/A

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

      7. lift-/.f32 N/A

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

      8. clear-num N/A

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

      9. lift-*.f32 N/A

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

      10. associate-/r* N/A

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

      11. div-inv N/A

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

      12. metadata-eval N/A

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

      13. lift-*.f32 N/A

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

      14. times-frac N/A

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

      15. lower-*.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. lower-/.f32 85.5

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

   10. Applied rewrites 85.5%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot s}{x} \cdot \frac{2}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 20000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 20000.0) 0.5 (* 2.0 (/ (* s s) (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 20000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 2.0f * ((s * s) / (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 20000.0e0) then
        tmp = 0.5e0
    else
        tmp = 2.0e0 * ((s * s) / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(20000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(s * s) / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(20000.0))
		tmp = single(0.5);
	else
		tmp = single(2.0) * ((s * s) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 2e4

   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 50.3%

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

   if 2e4 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)}} \]
   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 80.4%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 83.9

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

   8. Applied rewrites 83.9%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. /-rgt-identity N/A

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

      6. /-rgt-identity N/A

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

      7. lift-/.f32 N/A

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

      8. clear-num N/A

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

      9. frac-2neg N/A

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

      10. neg-mul-1 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. times-frac N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f32 N/A

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

      19. lower-/.f32 82.9

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

   10. Applied rewrites 82.9%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 20000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 999999995904:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(s \cdot s\right) \cdot \frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 999999995904.0) 0.5 (* (* s s) (/ 2.0 (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 999999995904.0f) {
		tmp = 0.5f;
	} else {
		tmp = (s * s) * (2.0f / (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 999999995904.0e0) then
        tmp = 0.5e0
    else
        tmp = (s * s) * (2.0e0 / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(999999995904.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(s * s) * Float32(Float32(2.0) / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(999999995904.0))
		tmp = single(0.5);
	else
		tmp = (s * s) * (single(2.0) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 999999996000

   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 47.2%

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

   if 999999996000 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)}} \]
   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 92.6%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 94.7

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

   8. Applied rewrites 94.7%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. /-rgt-identity N/A

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

      6. /-rgt-identity N/A

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

      7. lift-/.f32 N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 93.5

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

   10. Applied rewrites 93.5%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 999999995904:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(s \cdot s\right) \cdot \frac{2}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 500000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(2 \cdot \frac{s}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 500000.0) 0.5 (* s (* 2.0 (/ s (* x x))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 500000.0f) {
		tmp = 0.5f;
	} else {
		tmp = s * (2.0f * (s / (x * x)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 500000.0e0) then
        tmp = 0.5e0
    else
        tmp = s * (2.0e0 * (s / (x * x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(500000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(s * Float32(Float32(2.0) * Float32(s / Float32(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(500000.0))
		tmp = single(0.5);
	else
		tmp = s * (single(2.0) * (s / (x * x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5e5

   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 49.6%

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

   if 5e5 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)}} \]
   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 82.9%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 85.5

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

   8. Applied rewrites 85.5%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. /-rgt-identity N/A

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

      6. /-rgt-identity N/A

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

      7. lift-/.f32 N/A

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

      8. clear-num N/A

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

      9. lift-*.f32 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f32 N/A

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

      12. remove-double-neg N/A

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

      13. neg-mul-1 N/A

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

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

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

      15. metadata-eval N/A

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

      16. lift-*.f32 N/A

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

      17. times-frac N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f32 N/A

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

      20. lower-/.f32 80.4

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

   10. Applied rewrites 80.4%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 500000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(2 \cdot \frac{s}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.2% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) -5.0) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= -5.0f) {
		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) <= (-5.0e0)) 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(-5.0))
		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(-5.0))
		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) < -5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

   if -5 < (/.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 67.7

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

   5. Applied rewrites 67.7%

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

4. Final simplification 52.6%

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

Alternative 10: 47.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 0.5f) {
		tmp = 0.5f;
	} else {
		tmp = -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) <= 0.5e0) then
        tmp = 0.5e0
    else
        tmp = (-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(0.5))
		tmp = Float32(0.5);
	else
		tmp = 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(0.5))
		tmp = single(0.5);
	else
		tmp = single(-1.0) / (x / 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 51.3%

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

   if 0.5 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f32 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f32 46.8

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

   8. Applied rewrites 46.8%

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

      1. lift-neg.f32 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. lift-neg.f32 N/A

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

      6. frac-2neg N/A

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

      7. lower-*.f32 N/A

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

      8. lower-/.f32 46.8

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

   10. Applied rewrites 46.8%

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

       1. associate-*l/ N/A

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

       2. associate-/l* N/A

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

       3. clear-num N/A

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

       4. lift-/.f32 N/A

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

       5. un-div-inv N/A

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

       6. lower-/.f32 49.8

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

   12. Applied rewrites 49.8%

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

4. Final simplification 50.7%

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

Alternative 11: 46.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(0.5))
		tmp = Float32(0.5);
	else
		tmp = Float32(s * Float32(Float32(-1.0) / x));
	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 = s * (single(-1.0) / x);
	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 51.3%

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

   if 0.5 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f32 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f32 46.8

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

   8. Applied rewrites 46.8%

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

      1. lift-neg.f32 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. lift-neg.f32 N/A

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

      6. frac-2neg N/A

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

      7. lower-*.f32 N/A

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

      8. lower-/.f32 46.8

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

   10. Applied rewrites 46.8%

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

4. Final simplification 49.6%

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

Alternative 12: 46.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(0.5))
		tmp = Float32(0.5);
	else
		tmp = Float32(-Float32(s / x));
	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 = -(s / x);
	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 51.3%

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

   if 0.5 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f32 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f32 46.8

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

   8. Applied rewrites 46.8%

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

4. Final simplification 49.6%

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

Alternative 13: 34.2% 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 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 35.0%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))