Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 10.1s

Alternatives: 14

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. Final simplification 99.8%

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

Alternative 2: 49.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (- (/ x s))) 1.9999999494757503e-5) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (expf(-(x / s)) <= 1.9999999494757503e-5f) {
		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)) <= 1.9999999494757503e-5) 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(1.9999999494757503e-5))
		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(1.9999999494757503e-5))
		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)) < 1.99999995e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 28.1%

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

   if 1.99999995e-5 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 61.0

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

   5. Simplified 61.0%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (expf(-(x / s)) <= 2.0f) {
		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 (exp(-(x / s)) <= 2.0e0) 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 (exp(Float32(-Float32(x / s))) <= Float32(2.0))
		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 (exp(-(x / s)) <= single(2.0))
		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 (exp.f32 (/.f32 (neg.f32 x) s)) < 2

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 51.1%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 42.4

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

   5. Simplified 42.4%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f32 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f32 39.2

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

   8. Simplified 39.2%

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

      1. lift-neg.f32 N/A

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

      2. clear-num N/A

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

      3. metadata-eval N/A

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

      4. lift-neg.f32 N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-/.f32 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f32 42.4

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

   10. Applied egg-rr 42.4%

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

4. Final simplification 47.5%

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

Alternative 4: 47.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp(-(x / s)) <= single(2.0))
		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)) < 2

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 51.1%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 42.4

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

   5. Simplified 42.4%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f32 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f32 39.2

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

   8. Simplified 39.2%

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

4. Final simplification 46.2%

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

Alternative 5: 66.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \left(x \cdot -0.16666666666666666\right) \cdot \frac{x}{s \cdot \left(s \cdot s\right)}, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 0.20000000298023224)
   0.5
   (/ 1.0 (fma x (* (* x -0.16666666666666666) (/ x (* s (* s s)))) 2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 0.20000000298023224f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, ((x * -0.16666666666666666f) * (x / (s * (s * s)))), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(0.20000000298023224))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(Float32(x * Float32(-0.16666666666666666)) * Float32(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) < 0.200000003

   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 51.1%

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

   if 0.200000003 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Simplified 88.5%

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

   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. lower-/.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 88.0

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

   8. Simplified 88.0%

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

      1. associate-*r* N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-/.f32 91.9

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

   10. Applied egg-rr 91.9%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \left(x \cdot -0.16666666666666666\right) \cdot \frac{x}{s \cdot \left(s \cdot s\right)}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 200:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 200.0)
   0.5
   (/ 1.0 (* x (* (* x x) (/ -0.16666666666666666 (* s (* s s))))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 200.0f) {
		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) <= 200.0e0) 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(200.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(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(200.0))
		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) < 200

   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. Simplified 49.8%

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

   if 200 < (/.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. lower-fma.f32 N/A

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

   5. Simplified 91.2%

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

   6. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. sub-neg N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. distribute-neg-frac N/A

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

   8. Simplified 92.7%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f32 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 92.7

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

   11. Simplified 92.7%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 200:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x \cdot 0.5}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 0.20000000298023224)
   0.5
   (/ 1.0 (* x (/ (* x 0.5) (* s s))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 0.20000000298023224f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (x * ((x * 0.5f) / (s * s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 0.20000000298023224e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (x * ((x * 0.5e0) / (s * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(0.20000000298023224))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(x * Float32(0.5)) / Float32(s * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(0.20000000298023224))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (x * ((x * single(0.5)) / (s * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.200000003

   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 51.1%

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

   if 0.200000003 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Simplified 88.5%

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

   6. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. sub-neg N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. distribute-neg-frac N/A

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

   8. Simplified 88.8%

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

   9. Taylor expanded in x around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f32 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 82.5

          \[\leadsto \frac{1}{x \cdot \frac{x \cdot 0.5}{\color{blue}{s \cdot s}}} \]

   11. Simplified 82.5%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x \cdot 0.5}{s \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 2900:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(\left(s \cdot s\right) \cdot -6\right)}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 2900.0) 0.5 (/ (* s (* (* s s) -6.0)) (* x (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 2900.0f) {
		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) <= 2900.0e0) 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(2900.0))
		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(2900.0))
		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) < 2900

   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. Simplified 48.7%

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

   if 2900 < (/.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. lower-fma.f32 N/A

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

   5. Simplified 90.8%

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

   6. 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. lower-/.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 94.6

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

   8. Simplified 94.6%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.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. lower-*.f32 N/A

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

       8. lower-*.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. lower-*.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. lower-*.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. lower-*.f32 88.4

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

   11. Simplified 88.4%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 2900:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(\left(s \cdot s\right) \cdot -6\right)}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 2900:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(s \cdot \left(s \cdot s\right)\right) \cdot \frac{-6}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 2900.0) 0.5 (* (* s (* s s)) (/ -6.0 (* x (* x x))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 2900.0f) {
		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) <= 2900.0e0) 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(2900.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(s * Float32(s * s)) * Float32(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(2900.0))
		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) < 2900

   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. Simplified 48.7%

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

   if 2900 < (/.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. lower-fma.f32 N/A

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

   5. Simplified 90.8%

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

   6. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. sub-neg N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. distribute-neg-frac N/A

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

   8. Simplified 95.5%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. metadata-eval N/A

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

       5. distribute-neg-frac N/A

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

       6. metadata-eval N/A

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

       7. associate-*r/ N/A

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

       8. lower-*.f32 N/A

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

       11. lower-*.f32 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f32 N/A

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

       14. associate-*r/ N/A

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

       15. metadata-eval N/A

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

       16. distribute-neg-frac N/A

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

       17. metadata-eval N/A

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

       18. lower-/.f32 N/A

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

       19. cube-mult N/A

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

       20. unpow2 N/A

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

       21. lower-*.f32 N/A

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

       22. unpow2 N/A

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

       23. lower-*.f32 85.3

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

   11. Simplified 85.3%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 2900:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(s \cdot \left(s \cdot s\right)\right) \cdot \frac{-6}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 5.0000000900125474 \cdot 10^{-36}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, 0.5, -s\right)}{s \cdot s}, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 5.0000000900125474e-36)
   0.5
   (/ 1.0 (fma x (/ (fma x 0.5 (- s)) (* s s)) 2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 5.0000000900125474e-36f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, (fmaf(x, 0.5f, -s) / (s * s)), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(5.0000000900125474e-36))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(fma(x, Float32(0.5), Float32(-s)) / Float32(s * s)), Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 5.00000009e-36

   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 43.8%

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

   if 5.00000009e-36 < (neg.f32 x)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. 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. Taylor expanded in x around 0

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

   8. Simplified 84.3%

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

   9. Taylor expanded in s around 0

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

       1. lower-/.f32 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f32 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f32 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f32 84.3

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

   11. Simplified 84.3%

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

Alternative 11: 61.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 200:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(s \cdot 2\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 200.0) 0.5 (/ (* s (* s 2.0)) (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 200.0f) {
		tmp = 0.5f;
	} else {
		tmp = (s * (s * 2.0f)) / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 200.0e0) then
        tmp = 0.5e0
    else
        tmp = (s * (s * 2.0e0)) / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(200.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(s * Float32(s * Float32(2.0))) / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(200.0))
		tmp = single(0.5);
	else
		tmp = (s * (s * single(2.0))) / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 200

   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. Simplified 49.8%

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

   if 200 < (/.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. lower-fma.f32 N/A

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

   5. Simplified 91.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 84.9%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f32 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f32 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f32 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f32 79.7

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

   11. Simplified 79.7%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 200:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(s \cdot 2\right)}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 200:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot s\right) \cdot 2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 200.0) 0.5 (/ (* (* s s) 2.0) (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 200.0f) {
		tmp = 0.5f;
	} else {
		tmp = ((s * s) * 2.0f) / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 200.0e0) then
        tmp = 0.5e0
    else
        tmp = ((s * s) * 2.0e0) / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(200.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(s * s) * Float32(2.0)) / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(200.0))
		tmp = single(0.5);
	else
		tmp = ((s * s) * single(2.0)) / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 200

   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. Simplified 49.8%

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

   if 200 < (/.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. lower-fma.f32 N/A

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

   5. Simplified 91.2%

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

   6. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. sub-neg N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. distribute-neg-frac N/A

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

   8. Simplified 92.7%

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{0.5}{x \cdot \left(s \cdot s\right)} + \frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)}\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-/.f32 N/A

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

      8. lift-+.f32 N/A

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

      9. associate-*r* N/A

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

      10. lift-+.f32 N/A

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

      11. lift-/.f32 N/A

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

      12. lift-/.f32 N/A

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

      13. frac-add N/A

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

   10. Applied egg-rr 40.9%

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

   11. Taylor expanded in x around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f32 N/A

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

       3. lower-*.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. lower-*.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. lower-*.f32 79.7

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

   13. Simplified 79.7%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 200:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot s\right) \cdot 2}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 2900:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{s \cdot -2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 2900.0) 0.5 (* s (/ (* s -2.0) (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 2900.0f) {
		tmp = 0.5f;
	} else {
		tmp = s * ((s * -2.0f) / (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 2900.0e0) then
        tmp = 0.5e0
    else
        tmp = s * ((s * (-2.0e0)) / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(2900.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(s * Float32(Float32(s * Float32(-2.0)) / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(2900.0))
		tmp = single(0.5);
	else
		tmp = s * ((s * single(-2.0)) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 2900

   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. Simplified 48.7%

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

   if 2900 < (/.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 45.4

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

   5. Simplified 45.4%

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

   6. Taylor expanded in s around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f32 42.0

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

   8. Simplified 42.0%

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

   9. Taylor expanded in s around inf

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

       1. associate-*r/ N/A

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

       2. lower-/.f32 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 73.3

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

   11. Simplified 73.3%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 2900:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{s \cdot -2}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 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 99.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 32.7%

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

Reproduce

herbie shell --seed 2024208 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))