Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

Alternatives: 11

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.6000000238418579:\\ \;\;\;\;\frac{1}{\frac{s - x}{s} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ (exp (/ (- x) s)) 1.0)) 0.6000000238418579)
   (/ 1.0 (+ (/ (- s x) s) 1.0))
   (/ 1.0 (fma (- 1.0 (/ x s)) 1.0 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (expf((-x / s)) + 1.0f)) <= 0.6000000238418579f) {
		tmp = 1.0f / (((s - x) / s) + 1.0f);
	} else {
		tmp = 1.0f / fmaf((1.0f - (x / s)), 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(exp(Float32(Float32(-x) / s)) + Float32(1.0))) <= Float32(0.6000000238418579))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(s - x) / s) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(1.0) - Float32(x / s)), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.600000024

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in s around inf

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

      1. *-inverses N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 55.3

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

   8. Applied rewrites 55.3%

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

   if 0.600000024 < (/.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. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.0

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

   7. Applied rewrites 98.1%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.6000000238418579:\\ \;\;\;\;\frac{1}{\frac{s - x}{s} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.6000000238418579:\\ \;\;\;\;\frac{1}{\frac{s - x}{s} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1 - \frac{x}{s}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ (exp (/ (- x) s)) 1.0)) 0.6000000238418579)
   (/ 1.0 (+ (/ (- s x) s) 1.0))
   (/ 1.0 (fma 1.0 (- 1.0 (/ x s)) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (expf((-x / s)) + 1.0f)) <= 0.6000000238418579f) {
		tmp = 1.0f / (((s - x) / s) + 1.0f);
	} else {
		tmp = 1.0f / fmaf(1.0f, (1.0f - (x / s)), 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(exp(Float32(Float32(-x) / s)) + Float32(1.0))) <= Float32(0.6000000238418579))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(s - x) / s) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), Float32(Float32(1.0) - Float32(x / 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)))) < 0.600000024

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in s around inf

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

      1. *-inverses N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 55.3

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

   8. Applied rewrites 55.3%

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

   if 0.600000024 < (/.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. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 28.4%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.0

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

   9. Applied rewrites 99.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.6000000238418579:\\ \;\;\;\;\frac{1}{\frac{s - x}{s} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1 - \frac{x}{s}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 0.10000000149011612:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(0.5 \cdot x - s\right) \cdot x}{s \cdot s} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0)
     (/ 1.0 (fma (- 1.0 (/ x s)) 1.0 1.0))
     (if (<= t_0 0.10000000149011612)
       (+ (* 0.25 (/ x s)) 0.5)
       (/ 1.0 (+ (/ (* (- (* 0.5 x) s) x) (* s s)) 1.0))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f / fmaf((1.0f - (x / s)), 1.0f, 1.0f);
	} else if (t_0 <= 0.10000000149011612f) {
		tmp = (0.25f * (x / s)) + 0.5f;
	} else {
		tmp = 1.0f / (((((0.5f * x) - s) * x) / (s * s)) + 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(1.0) - Float32(x / s)), Float32(1.0), Float32(1.0)));
	elseif (t_0 <= Float32(0.10000000149011612))
		tmp = Float32(Float32(Float32(0.25) * Float32(x / s)) + Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(Float32(0.5) * x) - s) * x) / Float32(s * s)) + Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.0

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

   7. Applied rewrites 98.1%

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

   if -2 < (/.f32 (neg.f32 x) s) < 0.100000001

   1. Initial program 99.7%

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

   3. Taylor expanded in s around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{x}{s} + \frac{1}{2}} \]

      2. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{4}, \frac{x}{s}, \frac{1}{2}\right)} \]

      3. lower-/.f32 87.9

         \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\frac{x}{s}}, 0.5\right) \]

   5. Applied rewrites 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{x}{s}, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.2%

      \[\leadsto \frac{x}{s} \cdot 0.25 + \color{blue}{0.5} \]

   if 0.100000001 < (/.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

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 81.1%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \mathbf{elif}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(0.5 \cdot x - s\right) \cdot x}{s \cdot s} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.0

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

   7. Applied rewrites 98.1%

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

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

   1. Initial program 99.8%

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

      1. remove-double-div N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lift-+.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 99.8

          \[\leadsto \frac{1}{{\left({\left(e^{\frac{-x}{s}} + 1\right)}^{-0.5}\right)}^{\color{blue}{-2}}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \frac{1}{\color{blue}{{\left({\left(e^{\frac{-x}{s}} + 1\right)}^{-0.5}\right)}^{-2}}} \]

   5. Taylor expanded in s around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-/.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. lower-exp.f32 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f32 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

   10. Applied rewrites 82.9%

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

4. Final simplification 89.8%

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

Alternative 6: 47.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 28.1%

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

   9. Applied rewrites 28.3%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in s around inf

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

      1. *-inverses N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 55.3

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

   8. Applied rewrites 55.3%

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

4. Final simplification 43.6%

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

Alternative 7: 47.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.1

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

   5. Applied rewrites 5.1%

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

   7. Applied rewrites 28.1%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 28.4%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in s around inf

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

      1. *-inverses N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 55.3

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

   8. Applied rewrites 55.3%

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

4. Final simplification 43.6%

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

Alternative 8: 49.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 28.1%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in s around inf

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

      1. *-inverses N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 55.3

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

   8. Applied rewrites 55.3%

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

4. Final simplification 43.6%

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

Alternative 9: 49.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 28.1%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 55.3

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

   5. Applied rewrites 55.3%

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

4. Final simplification 43.6%

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

Alternative 10: 49.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 28.1%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 55.3

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

   5. Applied rewrites 55.3%

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

Alternative 11: 34.9% accurate, 128.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 33.2%

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

Reproduce

herbie shell --seed 2024271 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))