Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 9.7s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 49.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.0020000000949949026:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 0.0020000000949949026) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 0.0020000000949949026f) {
		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 (exp((-x / s)) <= 0.0020000000949949026e0) 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 (exp(Float32(Float32(-x) / s)) <= Float32(0.0020000000949949026))
		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 (exp((-x / s)) <= single(0.0020000000949949026))
		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 (exp.f32 (/.f32 (neg.f32 x) s)) < 0.00200000009

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

   if 0.00200000009 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 62.8

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

   5. Applied rewrites 62.8%

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

Alternative 3: 48.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;e^{t\_0} \leq 10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s))) (if (<= (exp t_0) 10.0) 0.5 (/ 1.0 t_0))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (expf(t_0) <= 10.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (exp(t_0) <= 10.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (exp(t_0) <= Float32(10.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (exp(t_0) <= single(10.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.4%

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

   if 10 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 40.4

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 40.4%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 50:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \mathsf{fma}\left(x, \frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)}, \frac{0.5}{s \cdot s}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 50.0)
   0.5
   (/
    1.0
    (*
     x
     (* x (fma x (/ -0.16666666666666666 (* s (* s s))) (/ 0.5 (* s s))))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 50.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (x * (x * fmaf(x, (-0.16666666666666666f / (s * (s * s))), (0.5f / (s * s)))));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(50.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(x * fma(x, Float32(Float32(-0.16666666666666666) / Float32(s * Float32(s * s))), Float32(Float32(0.5) / Float32(s * s))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.6%

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

   if 50 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 41.2

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

   5. Applied rewrites 41.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   8. Applied rewrites 92.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 85.2%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 93.6%

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

Alternative 5: 65.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 500:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(0.5, x \cdot s, x \cdot \left(x \cdot -0.16666666666666666\right)\right)}{s \cdot \left(s \cdot s\right)}, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 500.0)
   0.5
   (/
    1.0
    (fma
     x
     (/ (fma 0.5 (* x s) (* x (* x -0.16666666666666666))) (* s (* s s)))
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 500.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, (fmaf(0.5f, (x * s), (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(500.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(fma(Float32(0.5), Float32(x * s), Float32(x * 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) < 500

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 54.7%

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

   if 500 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 42.3

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

   5. Applied rewrites 42.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   8. Applied rewrites 94.4%

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

   9. Taylor expanded in s around 0

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

   11. Applied rewrites 91.2%

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

Alternative 6: 65.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, \mathsf{fma}\left(0.5, x, -s\right), x \cdot \left(x \cdot -0.16666666666666666\right)\right)}{s \cdot \left(s \cdot s\right)}, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 9.999999998199587e-24)
   0.5
   (/
    1.0
    (fma
     x
     (/
      (fma s (fma 0.5 x (- s)) (* x (* x -0.16666666666666666)))
      (* s (* s s)))
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 9.999999998199587e-24f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, (fmaf(s, fmaf(0.5f, x, -s), (x * (x * -0.16666666666666666f))) / (s * (s * s))), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(9.999999998199587e-24))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(fma(s, fma(Float32(0.5), x, Float32(-s)), Float32(x * 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 (neg.f32 x) < 1e-23

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 52.4%

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

   if 1e-23 < (neg.f32 x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   8. Applied rewrites 90.4%

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

   9. Taylor expanded in s around 0

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

   11. Applied rewrites 90.4%

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

Alternative 7: 65.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 500:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x \cdot \left(x \cdot -0.16666666666666666\right)}{s \cdot \left(s \cdot s\right)}, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 500.0)
   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) <= 500.0f) {
		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(500.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(Float32(x * 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) < 500

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 54.7%

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

   if 500 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 42.3

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

   5. Applied rewrites 42.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   8. Applied rewrites 94.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 91.2%

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

Alternative 8: 63.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 5000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 5000.0)
   0.5
   (/ 1.0 (* (* x (* x x)) (/ -0.16666666666666666 (* s (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 5000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((x * (x * x)) * (-0.16666666666666666f / (s * (s * s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 5000.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((x * (x * x)) * ((-0.16666666666666666e0) / (s * (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(5000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(x * x)) * Float32(Float32(-0.16666666666666666) / Float32(s * Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(5000.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((x * (x * x)) * (single(-0.16666666666666666) / (s * (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5e3

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 54.4%

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

   if 5e3 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 42.7

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

   5. Applied rewrites 42.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   8. Applied rewrites 94.3%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 88.8%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 88.8%

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

Alternative 9: 63.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 5000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{s \cdot \left(s \cdot s\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 5000.0)
   0.5
   (/ 1.0 (/ (* -0.16666666666666666 (* x (* x x))) (* s (* s s))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 5000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((-0.16666666666666666f * (x * (x * x))) / (s * (s * s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 5000.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (((-0.16666666666666666e0) * (x * (x * x))) / (s * (s * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(5000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(-0.16666666666666666) * Float32(x * Float32(x * x))) / Float32(s * Float32(s * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(5000.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((single(-0.16666666666666666) * (x * (x * x))) / (s * (s * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5e3

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 54.4%

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

   if 5e3 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

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

      1. lower-+.f32 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f32 N/A

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 86.7%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 5000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{s \cdot \left(s \cdot s\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-5.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(fma(Float32(0.5), Float32(x / s), Float32(-1.0)) / s), Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

   if -5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Applied rewrites 81.1%

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

   7. Applied rewrites 83.6%

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

Alternative 11: 63.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.1599999964237213:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(0.5, x, -s\right)}{s \cdot s}, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 0.1599999964237213)
   0.5
   (/ 1.0 (fma x (/ (fma 0.5 x (- s)) (* s s)) 2.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.1599999964237213f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, (fmaf(0.5f, x, -s) / (s * s)), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(0.1599999964237213))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(fma(Float32(0.5), x, 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) < 0.159999996

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.9%

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

   if 0.159999996 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Applied rewrites 69.6%

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

   7. Applied rewrites 73.9%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 78.9%

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

Alternative 12: 63.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.1%

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

   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

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Applied rewrites 70.9%

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 81.0%

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

4. Final simplification 65.6%

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

Alternative 13: 63.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 10.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (x * (0.5e0 * (x / (s * s))))
    end if
    code = tmp
end function

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) / Float32(x * Float32(Float32(0.5) * Float32(x / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(10.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (x * (single(0.5) * (x / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 56.1%

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

   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

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 76.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 81.0%

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

Alternative 14: 49.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= (-5.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 + ((-1.0e0) / (s / x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-5.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(-1.0) / Float32(s / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-5.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) + (single(-1.0) / (s / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

   if -5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 62.8

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

   5. Applied rewrites 62.8%

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

   7. Applied rewrites 62.8%

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

4. Final simplification 51.7%

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

Alternative 15: 35.7% 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 99.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 37.1%

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

Reproduce

herbie shell --seed 2024233 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))