Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 11.3s

Alternatives: 17

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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (exp (- (log1p (exp (- (/ x s)))))))

C

float code(float x, float s) {
	return expf(-log1pf(expf(-(x / s))));
}

Julia

function code(x, s)
	return exp(Float32(-log1p(exp(Float32(-Float32(x / s))))))
end

Derivation

1. Initial program 99.9%

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

   1. inv-pow N/A

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

   2. pow-to-exp N/A

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

   3. *-commutative N/A

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

   4. log-pow N/A

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

   5. inv-pow N/A

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

   6. exp-lowering-exp.f32 N/A

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

   7. log-rec N/A

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

   8. neg-lowering-neg.f32 N/A

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

   9. accelerator-lowering-log1p.f32 N/A

      \[\leadsto e^{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)}\right)} \]

   10. exp-lowering-exp.f32 N/A

       \[\leadsto e^{\mathsf{neg}\left(\mathsf{log1p}\left(\color{blue}{e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right)} \]

   11. distribute-frac-neg N/A

       \[\leadsto e^{\mathsf{neg}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{neg}\left(\frac{x}{s}\right)}}\right)\right)} \]

   12. neg-lowering-neg.f32 N/A

       \[\leadsto e^{\mathsf{neg}\left(\mathsf{log1p}\left(e^{\color{blue}{\mathsf{neg}\left(\frac{x}{s}\right)}}\right)\right)} \]

   13. /-lowering-/.f32 99.9

       \[\leadsto e^{-\mathsf{log1p}\left(e^{-\color{blue}{\frac{x}{s}}}\right)} \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)}} \]
5. Add Preprocessing

Alternative 2: 47.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (- (/ x s))) 4.0) 0.5 (* s (/ -1.0 x))))

C

float code(float x, float s) {
	float tmp;
	if (expf(-(x / s)) <= 4.0f) {
		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 (exp(-(x / s)) <= 4.0e0) 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 (exp(Float32(-Float32(x / s))) <= Float32(4.0))
		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 (exp(-(x / s)) <= single(4.0))
		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 (exp.f32 (/.f32 (neg.f32 x) s)) < 4

   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. Simplified 55.5%

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

   if 4 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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. --lowering--.f32 N/A

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

      4. /-lowering-/.f32 39.1

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

   5. Simplified 39.1%

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

   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. /-lowering-/.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. neg-lowering-neg.f32 33.9

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

   8. Simplified 33.9%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f32 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. frac-2neg N/A

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

      6. /-lowering-/.f32 33.9

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

   10. Applied egg-rr 33.9%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (- (/ x s))) 4.0) 0.5 (/ s (- x))))

C

float code(float x, float s) {
	float tmp;
	if (expf(-(x / s)) <= 4.0f) {
		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 (exp(-(x / s)) <= 4.0e0) then
        tmp = 0.5e0
    else
        tmp = s / -x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(-Float32(x / s))) <= Float32(4.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(s / Float32(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp(-(x / s)) <= single(4.0))
		tmp = single(0.5);
	else
		tmp = s / -x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 4

   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. Simplified 55.5%

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

   if 4 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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. --lowering--.f32 N/A

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

      4. /-lowering-/.f32 39.1

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

   5. Simplified 39.1%

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

   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. /-lowering-/.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. neg-lowering-neg.f32 33.9

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

   8. Simplified 33.9%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 5: 68.2% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) -10.0)
   0.5
   (/
    1.0
    (fma
     x
     (fma (/ (fma (/ x s) -0.16666666666666666 0.5) s) (/ x s) (/ -1.0 s))
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= -10.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, fmaf((fmaf((x / s), -0.16666666666666666f, 0.5f) / s), (x / s), (-1.0f / s)), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(-10.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, fma(Float32(fma(Float32(x / s), Float32(-0.16666666666666666), Float32(0.5)) / s), Float32(x / s), 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) < -10

   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. Simplified 28.1%

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

   if -10 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   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. accelerator-lowering-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. Simplified 89.2%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. /-lowering-/.f32 N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. /-lowering-/.f32 N/A

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

      9. /-lowering-/.f32 N/A

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

      10. /-lowering-/.f32 94.6

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

   7. Applied egg-rr 94.6%

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

4. Final simplification 72.3%

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

Alternative 6: 65.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{s}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 99999997952:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\frac{x}{s}, 0.5, -1\right)}{s}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (- (/ x s))))
   (if (<= t_0 -10.0)
     0.5
     (if (<= t_0 99999997952.0)
       (/ 1.0 (fma x (/ (fma (/ x s) 0.5 -1.0) s) 2.0))
       (/ (* (* s (* s s)) -6.0) (* x (* x x)))))))

C

float code(float x, float s) {
	float t_0 = -(x / s);
	float tmp;
	if (t_0 <= -10.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 99999997952.0f) {
		tmp = 1.0f / fmaf(x, (fmaf((x / s), 0.5f, -1.0f) / s), 2.0f);
	} else {
		tmp = ((s * (s * s)) * -6.0f) / (x * (x * x));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(-Float32(x / s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-10.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(99999997952.0))
		tmp = Float32(Float32(1.0) / fma(x, Float32(fma(Float32(x / s), Float32(0.5), Float32(-1.0)) / s), Float32(2.0)));
	else
		tmp = Float32(Float32(Float32(s * Float32(s * s)) * Float32(-6.0)) / Float32(x * Float32(x * x)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -10

   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. Simplified 28.1%

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

   if -10 < (/.f32 (neg.f32 x) s) < 99999998000

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. 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. accelerator-lowering-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. Simplified 82.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. /-lowering-/.f32 88.2

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

   7. Applied egg-rr 88.2%

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

   if 99999998000 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   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. accelerator-lowering-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. Simplified 99.0%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. /-lowering-/.f32 N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. /-lowering-/.f32 N/A

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

      9. /-lowering-/.f32 N/A

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

      10. /-lowering-/.f32 99.0

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f32 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f32 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f32 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 N/A

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

      10. cube-mult N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

      11. unpow2 N/A

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

      12. *-lowering-*.f32 N/A

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

      13. unpow2 N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

      14. *-lowering-*.f32 96.3

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

   10. Simplified 96.3%

       \[\leadsto \color{blue}{\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-\frac{x}{s} \leq 99999997952:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\frac{x}{s}, 0.5, -1\right)}{s}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.2000000072537151 \cdot 10^{-35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(x, -0.16666666666666666 \cdot \frac{1}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.2000000072537151e-35)
   0.5
   (/
    1.0
    (fma
     x
     (fma
      (/ x (* s s))
      (fma x (* -0.16666666666666666 (/ 1.0 s)) 0.5)
      (/ -1.0 s))
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.2000000072537151e-35f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, fmaf((x / (s * s)), fmaf(x, (-0.16666666666666666f * (1.0f / s)), 0.5f), (-1.0f / s)), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(1.2000000072537151e-35))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, fma(Float32(x / Float32(s * s)), fma(x, Float32(Float32(-0.16666666666666666) * Float32(Float32(1.0) / 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 (neg.f32 x) < 1.20000001e-35

   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. Simplified 47.2%

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

   if 1.20000001e-35 < (neg.f32 x)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   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. accelerator-lowering-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. Simplified 94.7%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. /-lowering-/.f32 94.7

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

   7. Applied egg-rr 94.7%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 1.2000000072537151 \cdot 10^{-35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(x, -0.16666666666666666 \cdot \frac{1}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.2000000072537151 \cdot 10^{-35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.2000000072537151e-35)
   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 <= 1.2000000072537151e-35f) {
		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(-x) <= Float32(1.2000000072537151e-35))
		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 (neg.f32 x) < 1.20000001e-35

   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. Simplified 47.2%

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

   if 1.20000001e-35 < (neg.f32 x)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   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. accelerator-lowering-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. Simplified 94.7%

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

Alternative 9: 67.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 0.5f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, (x * ((x * -0.16666666666666666f) / (s * (s * s)))), 2.0f);
	}
	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) / fma(x, Float32(x * Float32(Float32(x * Float32(-0.16666666666666666)) / Float32(s * Float32(s * s)))), Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 55.6%

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

   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

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   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. accelerator-lowering-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. Simplified 94.2%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f32 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f32 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f32 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f32 90.8

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

   8. Simplified 90.8%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. *-lowering-*.f32 N/A

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

      7. *-lowering-*.f32 95.8

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

   10. Applied egg-rr 95.8%

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

4. Final simplification 71.3%

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

Alternative 10: 66.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -1.500000029312222 \cdot 10^{-25}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\frac{x}{s}, 0.5, -1\right)}{s}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right) \cdot -18}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -1.500000029312222e-25)
   0.5
   (if (<= (- x) 2.0000000233721948e-7)
     (/ 1.0 (fma x (/ (fma (/ x s) 0.5 -1.0) s) 2.0))
     (/ (* (* (* s s) (* s s)) -18.0) (* x (* x (* x x)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -1.500000029312222e-25f) {
		tmp = 0.5f;
	} else if (-x <= 2.0000000233721948e-7f) {
		tmp = 1.0f / fmaf(x, (fmaf((x / s), 0.5f, -1.0f) / s), 2.0f);
	} else {
		tmp = (((s * s) * (s * s)) * -18.0f) / (x * (x * (x * x)));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-1.500000029312222e-25))
		tmp = Float32(0.5);
	elseif (Float32(-x) <= Float32(2.0000000233721948e-7))
		tmp = Float32(Float32(1.0) / fma(x, Float32(fma(Float32(x / s), Float32(0.5), Float32(-1.0)) / s), Float32(2.0)));
	else
		tmp = Float32(Float32(Float32(Float32(s * s) * Float32(s * s)) * Float32(-18.0)) / Float32(x * Float32(x * Float32(x * x))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < -1.50000003e-25

   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. Simplified 37.0%

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

   if -1.50000003e-25 < (neg.f32 x) < 2.00000002e-7

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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. accelerator-lowering-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. Simplified 76.9%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. /-lowering-/.f32 86.3

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

   7. Applied egg-rr 86.3%

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

   if 2.00000002e-7 < (neg.f32 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   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. accelerator-lowering-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. Simplified 94.8%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. /-lowering-/.f32 N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. /-lowering-/.f32 N/A

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

      9. /-lowering-/.f32 N/A

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

      10. /-lowering-/.f32 94.8

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

   7. Applied egg-rr 94.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-18 \cdot \frac{{s}^{4}}{x} + -6 \cdot {s}^{3}}{{x}^{3}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{-18 \cdot \frac{{s}^{4}}{x} + -6 \cdot {s}^{3}}{{x}^{3}}} \]

   10. Simplified 94.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(s, \left(s \cdot s\right) \cdot -6, \frac{\left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right) \cdot -18}{x}\right)}{x \cdot \left(x \cdot x\right)}} \]

   11. Taylor expanded in s around inf

       \[\leadsto \color{blue}{-18 \cdot \frac{{s}^{4}}{{x}^{4}}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{-18 \cdot {s}^{4}}{{x}^{4}}} \]

       2. /-lowering-/.f32 N/A

          \[\leadsto \color{blue}{\frac{-18 \cdot {s}^{4}}{{x}^{4}}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{{s}^{4} \cdot -18}}{{x}^{4}} \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \frac{\color{blue}{{s}^{4} \cdot -18}}{{x}^{4}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{{s}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot -18}{{x}^{4}} \]

       6. pow-sqr N/A

          \[\leadsto \frac{\color{blue}{\left({s}^{2} \cdot {s}^{2}\right)} \cdot -18}{{x}^{4}} \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \frac{\color{blue}{\left({s}^{2} \cdot {s}^{2}\right)} \cdot -18}{{x}^{4}} \]

       8. unpow2 N/A

          \[\leadsto \frac{\left(\color{blue}{\left(s \cdot s\right)} \cdot {s}^{2}\right) \cdot -18}{{x}^{4}} \]

       9. *-lowering-*.f32 N/A

          \[\leadsto \frac{\left(\color{blue}{\left(s \cdot s\right)} \cdot {s}^{2}\right) \cdot -18}{{x}^{4}} \]

       10. unpow2 N/A

           \[\leadsto \frac{\left(\left(s \cdot s\right) \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot -18}{{x}^{4}} \]

       11. *-lowering-*.f32 N/A

           \[\leadsto \frac{\left(\left(s \cdot s\right) \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot -18}{{x}^{4}} \]

       12. metadata-eval N/A

           \[\leadsto \frac{\left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right) \cdot -18}{{x}^{\color{blue}{\left(3 + 1\right)}}} \]

       13. pow-plus N/A

           \[\leadsto \frac{\left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right) \cdot -18}{\color{blue}{{x}^{3} \cdot x}} \]

       14. *-lowering-*.f32 N/A

           \[\leadsto \frac{\left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right) \cdot -18}{\color{blue}{{x}^{3} \cdot x}} \]

       15. cube-mult N/A

           \[\leadsto \frac{\left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right) \cdot -18}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot x} \]

       16. unpow2 N/A

           \[\leadsto \frac{\left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right) \cdot -18}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot x} \]

       17. *-lowering-*.f32 N/A

           \[\leadsto \frac{\left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right) \cdot -18}{\color{blue}{\left(x \cdot {x}^{2}\right)} \cdot x} \]

       18. unpow2 N/A

           \[\leadsto \frac{\left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right) \cdot -18}{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot x} \]

       19. *-lowering-*.f32 96.2

           \[\leadsto \frac{\left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right) \cdot -18}{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot x} \]

   13. Simplified 96.2%

       \[\leadsto \color{blue}{\frac{\left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right) \cdot -18}{\left(x \cdot \left(x \cdot x\right)\right) \cdot x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -1.500000029312222 \cdot 10^{-25}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\frac{x}{s}, 0.5, -1\right)}{s}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right) \cdot -18}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 1.0) 0.5 (/ (* (* s (* s s)) -6.0) (* x (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 1.0f) {
		tmp = 0.5f;
	} else {
		tmp = ((s * (s * s)) * -6.0f) / (x * (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 1.0e0) then
        tmp = 0.5e0
    else
        tmp = ((s * (s * s)) * (-6.0e0)) / (x * (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(1.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(s * Float32(s * s)) * Float32(-6.0)) / Float32(x * Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(1.0))
		tmp = single(0.5);
	else
		tmp = ((s * (s * s)) * single(-6.0)) / (x * (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   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. Simplified 55.5%

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

   if 1 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   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. accelerator-lowering-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. Simplified 94.6%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. times-frac N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. /-lowering-/.f32 N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. /-lowering-/.f32 N/A

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

      9. /-lowering-/.f32 N/A

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

      10. /-lowering-/.f32 92.8

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

   7. Applied egg-rr 92.8%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f32 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f32 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f32 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 N/A

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

      10. cube-mult N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

      11. unpow2 N/A

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

      12. *-lowering-*.f32 N/A

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

      13. unpow2 N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

      14. *-lowering-*.f32 86.0

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

   10. Simplified 86.0%

       \[\leadsto \color{blue}{\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(\left(s \cdot s\right) \cdot -6\right)}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 1.0) 0.5 (/ (* s (* (* s s) -6.0)) (* x (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 1.0f) {
		tmp = 0.5f;
	} else {
		tmp = (s * ((s * s) * -6.0f)) / (x * (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 1.0e0) then
        tmp = 0.5e0
    else
        tmp = (s * ((s * s) * (-6.0e0))) / (x * (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(1.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(s * Float32(Float32(s * s) * Float32(-6.0))) / Float32(x * Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(1.0))
		tmp = single(0.5);
	else
		tmp = (s * ((s * s) * single(-6.0))) / (x * (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   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. Simplified 55.5%

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

   if 1 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   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. accelerator-lowering-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. Simplified 94.6%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f32 N/A

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

      3. *-commutative N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f32 N/A

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

      8. *-lowering-*.f32 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f32 N/A

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

      11. cube-mult N/A

          \[\leadsto \frac{s \cdot \left(\left(s \cdot s\right) \cdot -6\right)}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

      12. unpow2 N/A

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

      13. *-lowering-*.f32 N/A

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

      14. unpow2 N/A

          \[\leadsto \frac{s \cdot \left(\left(s \cdot s\right) \cdot -6\right)}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

      15. *-lowering-*.f32 86.0

          \[\leadsto \frac{s \cdot \left(\left(s \cdot s\right) \cdot -6\right)}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

   8. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{s \cdot \left(\left(s \cdot s\right) \cdot -6\right)}{x \cdot \left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(\left(s \cdot s\right) \cdot -6\right)}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 1000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 1000.0) 0.5 (/ (* 2.0 (* s s)) (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 1000.0f) {
		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) <= 1000.0e0) 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(1000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(2.0) * 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(1000.0))
		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) < 1e3

   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. Simplified 54.8%

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

   if 1e3 < (/.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. accelerator-lowering-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. Simplified 72.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f32 N/A

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

      3. *-lowering-*.f32 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f32 77.2

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

   8. Simplified 77.2%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 1000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 500000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(s \cdot \frac{s}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 500000000.0) 0.5 (* 2.0 (* s (/ s (* x x))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 500000000.0f) {
		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) <= 500000000.0e0) 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(500000000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) * Float32(s * Float32(s / Float32(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(500000000.0))
		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) < 5e8

   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. Simplified 52.8%

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

   if 5e8 < (/.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. accelerator-lowering-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. Simplified 76.9%

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

   6. Taylor expanded in x around inf

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

      1. *-lowering-*.f32 N/A

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

      2. /-lowering-/.f32 76.9

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

   8. Simplified 76.9%

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

   9. Taylor expanded in x around inf

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

       1. *-lowering-*.f32 N/A

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

       2. unpow2 N/A

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

       3. associate-/l* N/A

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

       4. *-lowering-*.f32 N/A

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

       5. /-lowering-/.f32 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f32 75.3

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

   11. Simplified 75.3%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 500000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(s \cdot \frac{s}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= -1.0f) {
		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) <= (-1.0e0)) 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(-1.0))
		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(-1.0))
		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) < -1

   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. Simplified 28.2%

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

   if -1 < (/.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. --lowering--.f32 N/A

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

      4. /-lowering-/.f32 62.4

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

   5. Simplified 62.4%

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

4. Final simplification 50.8%

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

Alternative 16: 49.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 1.0f) {
		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) <= 1.0e0) 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(1.0))
		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(1.0))
		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) < 1

   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. Simplified 55.5%

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

   if 1 < (/.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. --lowering--.f32 N/A

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

      4. /-lowering-/.f32 39.1

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

   5. Simplified 39.1%

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

   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. /-lowering-/.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. neg-lowering-neg.f32 33.9

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

   8. Simplified 33.9%

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

      1. clear-num N/A

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

      2. metadata-eval N/A

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

      3. distribute-frac-neg N/A

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

      4. frac-2neg N/A

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

      5. /-lowering-/.f32 N/A

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

      6. /-lowering-/.f32 39.1

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

   10. Applied egg-rr 39.1%

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

4. Final simplification 49.1%

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

Alternative 17: 35.4% 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. Simplified 36.5%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))