Logistic function

Percentage Accurate: 99.8% → 99.6%

Time: 8.1s

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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \sqrt[3]{e^{\frac{-3}{\frac{s}{x}}}}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (cbrt (exp (/ -3.0 (/ s x)))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + cbrtf(expf((-3.0f / (s / x)))));
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + cbrt(exp(Float32(Float32(-3.0) / Float32(s / x))))))
end

Derivation

1. Initial program 99.8%

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

   1. div-inv 99.7%

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

   2. exp-prod 82.8%

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

   3. neg-mul-1 82.8%

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

   4. exp-prod 82.8%

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

   5. pow-pow 99.7%

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

   6. div-inv 99.8%

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

3. Applied egg-rr 99.8%

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

   1. add-cbrt-cube 99.8%

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

   2. pow3 99.8%

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

   3. pow-exp 99.8%

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

   4. pow-exp 99.8%

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

   5. neg-mul-1 99.8%

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

5. Applied egg-rr 99.8%

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

   1. distribute-lft-neg-out 99.8%

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

   2. neg-sub0 99.8%

      \[\leadsto \frac{1}{1 + \sqrt[3]{e^{\color{blue}{0 - \frac{x}{s} \cdot 3}}}} \]

   3. clear-num 99.7%

      \[\leadsto \frac{1}{1 + \sqrt[3]{e^{0 - \color{blue}{\frac{1}{\frac{s}{x}}} \cdot 3}}} \]

   4. associate-*l/ 99.8%

      \[\leadsto \frac{1}{1 + \sqrt[3]{e^{0 - \color{blue}{\frac{1 \cdot 3}{\frac{s}{x}}}}}} \]

   5. metadata-eval 99.8%

      \[\leadsto \frac{1}{1 + \sqrt[3]{e^{0 - \frac{\color{blue}{3}}{\frac{s}{x}}}}} \]

7. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \sqrt[3]{e^{\color{blue}{0 - \frac{3}{\frac{s}{x}}}}}} \]
8. Step-by-step derivation

   1. neg-sub0 99.8%

      \[\leadsto \frac{1}{1 + \sqrt[3]{e^{\color{blue}{-\frac{3}{\frac{s}{x}}}}}} \]

   2. distribute-neg-frac 99.8%

      \[\leadsto \frac{1}{1 + \sqrt[3]{e^{\color{blue}{\frac{-3}{\frac{s}{x}}}}}} \]

   3. metadata-eval 99.8%

      \[\leadsto \frac{1}{1 + \sqrt[3]{e^{\frac{\color{blue}{-3}}{\frac{s}{x}}}}} \]

9. Simplified 99.8%

   \[\leadsto \frac{1}{1 + \sqrt[3]{e^{\color{blue}{\frac{-3}{\frac{s}{x}}}}}} \]

10. Final simplification 99.8%

    \[\leadsto \frac{1}{1 + \sqrt[3]{e^{\frac{-3}{\frac{s}{x}}}}} \]

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf(-1.0f), (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((-1.0e0)) ** (x / s)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. div-inv 99.7%

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

   2. exp-prod 82.8%

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

   3. neg-mul-1 82.8%

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

   4. exp-prod 82.8%

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

   5. pow-pow 99.7%

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

   6. div-inv 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 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. Final simplification 99.8%

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

Alternative 4: 63.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.000000023742228 \cdot 10^{-32}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{x}{s}\right) + 0.5 \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.000000023742228e-32)
   0.5
   (/ 1.0 (+ (- 2.0 (/ x s)) (* 0.5 (* x (/ x (* s s))))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.000000023742228e-32f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((2.0f - (x / s)) + (0.5f * (x * (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 <= 1.000000023742228e-32) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((2.0e0 - (x / s)) + (0.5e0 * (x * (x / (s * s)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(1.000000023742228e-32))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(2.0) - Float32(x / s)) + Float32(Float32(0.5) * Float32(x * Float32(x / Float32(s * s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(1.000000023742228e-32))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((single(2.0) - (x / s)) + (single(0.5) * (x * (x / (s * s)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1.00000002e-32

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 48.4%

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

   if 1.00000002e-32 < (neg.f32 x)

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 78.4%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 78.4%

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

      2. associate-+l+ 78.4%

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

      3. unpow2 78.4%

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

      4. unpow2 78.4%

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

      5. times-frac 74.1%

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

      6. +-commutative 74.1%

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

      7. mul-1-neg 74.1%

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

      8. unsub-neg 74.1%

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

   4. Simplified 74.1%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) + \left(2 - \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. clear-num 74.1%

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

      2. frac-times 78.6%

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

      3. *-un-lft-identity 78.6%

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

   6. Applied egg-rr 78.6%

      \[\leadsto \frac{1}{0.5 \cdot \color{blue}{\frac{x}{\frac{s}{x} \cdot s}} + \left(2 - \frac{x}{s}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 83.9%

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

      2. associate-/r/ 83.9%

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

   8. Applied egg-rr 83.9%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 1.000000023742228 \cdot 10^{-32}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{x}{s}\right) + 0.5 \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}\\ \end{array} \]

Alternative 5: 63.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.000000023742228 \cdot 10^{-32}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{x}{\frac{s \cdot s}{x}} + \left(2 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.000000023742228e-32)
   0.5
   (/ 1.0 (+ (* 0.5 (/ x (/ (* s s) x))) (- 2.0 (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.000000023742228e-32f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((0.5f * (x / ((s * s) / x))) + (2.0f - (x / s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 1.000000023742228e-32) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((0.5e0 * (x / ((s * s) / x))) + (2.0e0 - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(1.000000023742228e-32))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(0.5) * Float32(x / Float32(Float32(s * s) / x))) + Float32(Float32(2.0) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(1.000000023742228e-32))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((single(0.5) * (x / ((s * s) / x))) + (single(2.0) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1.00000002e-32

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 48.4%

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

   if 1.00000002e-32 < (neg.f32 x)

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 78.4%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 78.4%

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

      2. associate-+l+ 78.4%

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

      3. unpow2 78.4%

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

      4. unpow2 78.4%

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

      5. times-frac 74.1%

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

      6. +-commutative 74.1%

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

      7. mul-1-neg 74.1%

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

      8. unsub-neg 74.1%

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

   4. Simplified 74.1%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) + \left(2 - \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. clear-num 74.1%

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

      2. frac-times 78.6%

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

      3. *-un-lft-identity 78.6%

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

   6. Applied egg-rr 78.6%

      \[\leadsto \frac{1}{0.5 \cdot \color{blue}{\frac{x}{\frac{s}{x} \cdot s}} + \left(2 - \frac{x}{s}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 83.9%

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

   8. Applied egg-rr 83.9%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 1.000000023742228 \cdot 10^{-32}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{x}{\frac{s \cdot s}{x}} + \left(2 - \frac{x}{s}\right)}\\ \end{array} \]

Alternative 6: 62.9% accurate, 6.7× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.5f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (0.5f * (x * (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) <= 0.5e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (0.5e0 * (x * (x / (s * s))))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 53.5%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 74.2%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 74.2%

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

      2. associate-+l+ 74.2%

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

      3. unpow2 74.2%

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

      4. unpow2 74.2%

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

      5. times-frac 68.4%

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

      6. +-commutative 68.4%

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

      7. mul-1-neg 68.4%

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

      8. unsub-neg 68.4%

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

   4. Simplified 68.4%

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

   5. Taylor expanded in x around inf 68.4%

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

      1. neg-mul-1 46.7%

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

      2. distribute-neg-frac 46.7%

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

   7. Simplified 68.4%

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

   8. Taylor expanded in x around inf 74.2%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{{x}^{2}}{{s}^{2}}}} \]
   9. Step-by-step derivation

      1. unpow2 74.2%

         \[\leadsto \frac{1}{0.5 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}} \]

      2. associate-*r/ 81.0%

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

      3. unpow2 81.0%

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

   10. Simplified 81.0%

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

4. Final simplification 63.2%

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

Alternative 7: 58.1% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 53.5%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 74.2%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 74.2%

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

      2. associate-+l+ 74.2%

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

      3. unpow2 74.2%

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

      4. unpow2 74.2%

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

      5. times-frac 68.4%

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

      6. +-commutative 68.4%

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

      7. mul-1-neg 68.4%

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

      8. unsub-neg 68.4%

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

   4. Simplified 68.4%

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

   5. Taylor expanded in x around inf 74.2%

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

      1. unpow2 74.2%

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

      2. unpow2 74.2%

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

      3. times-frac 68.3%

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

   7. Simplified 68.3%

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

4. Final simplification 58.7%

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

Alternative 8: 58.1% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 53.5%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 74.2%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 74.2%

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

      2. associate-+l+ 74.2%

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

      3. unpow2 74.2%

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

      4. unpow2 74.2%

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

      5. times-frac 68.4%

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

      6. +-commutative 68.4%

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

      7. mul-1-neg 68.4%

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

      8. unsub-neg 68.4%

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

   4. Simplified 68.4%

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

   5. Taylor expanded in x around inf 74.2%

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

      1. unpow2 74.2%

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

      2. unpow2 74.2%

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

      3. times-frac 68.3%

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

   7. Simplified 68.3%

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

      1. clear-num 68.3%

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

      2. frac-times 68.3%

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

      3. *-commutative 68.3%

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

      4. *-un-lft-identity 68.3%

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

   9. Applied egg-rr 68.3%

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

4. Final simplification 58.7%

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

Alternative 9: 59.6% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 53.5%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 74.2%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 74.2%

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

      2. associate-+l+ 74.2%

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

      3. unpow2 74.2%

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

      4. unpow2 74.2%

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

      5. times-frac 68.4%

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

      6. +-commutative 68.4%

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

      7. mul-1-neg 68.4%

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

      8. unsub-neg 68.4%

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

   4. Simplified 68.4%

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

   5. Taylor expanded in x around inf 74.2%

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

      1. unpow2 74.2%

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

      2. unpow2 74.2%

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

      3. times-frac 68.3%

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

   7. Simplified 68.3%

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

      1. *-commutative 68.3%

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

      2. frac-times 74.2%

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

      3. associate-*l/ 74.2%

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

   9. Applied egg-rr 74.2%

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

       1. times-frac 81.0%

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

       2. associate-/l* 74.1%

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

   11. Applied egg-rr 74.1%

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

4. Final simplification 60.8%

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

Alternative 10: 60.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 40000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot s\right) \cdot 2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 40000000.0) 0.5 (/ (* (* s s) 2.0) (* x x))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 40000000.0f) {
		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) <= 40000000.0e0) 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(40000000.0))
		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(40000000.0))
		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) < 4e7

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 49.6%

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

   if 4e7 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 88.5%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 88.5%

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

      2. associate-+l+ 88.5%

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

      3. unpow2 88.5%

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

      4. unpow2 88.5%

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

      5. times-frac 81.0%

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

      6. +-commutative 81.0%

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

      7. mul-1-neg 81.0%

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

      8. unsub-neg 81.0%

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

   4. Simplified 81.0%

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

   5. Taylor expanded in x around inf 88.5%

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

      1. unpow2 88.5%

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

      2. unpow2 88.5%

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

      3. times-frac 81.0%

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

   7. Simplified 81.0%

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

      1. *-commutative 81.0%

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

      2. frac-times 88.5%

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

      3. associate-*l/ 88.5%

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

   9. Applied egg-rr 88.5%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 40000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot s\right) \cdot 2}{x \cdot x}\\ \end{array} \]

Alternative 11: 48.7% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.2%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 67.8%

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

      1. mul-1-neg 67.8%

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

      2. unsub-neg 67.8%

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

   4. Simplified 67.8%

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

4. Final simplification 53.1%

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

Alternative 12: 47.2% accurate, 9.7× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 0.5f) {
		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 (t_0 <= 0.5e0) 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 (t_0 <= Float32(0.5))
		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 (t_0 <= single(0.5))
		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 (/.f32 (neg.f32 x) s) < 0.5

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 53.5%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 46.7%

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

      1. mul-1-neg 46.7%

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

      2. unsub-neg 46.7%

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

   4. Simplified 46.7%

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

   5. Taylor expanded in x around inf 46.7%

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

      1. neg-mul-1 46.7%

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

      2. distribute-neg-frac 46.7%

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

   7. Simplified 46.7%

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

4. Final simplification 51.1%

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

Alternative 13: 45.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 2.0000000233721948e-7) 0.5 (/ (- s) x)))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 2.0000000233721948e-7f) {
		tmp = 0.5f;
	} else {
		tmp = -s / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 2.0000000233721948e-7) then
        tmp = 0.5e0
    else
        tmp = -s / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(2.0000000233721948e-7))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(-s) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(2.0000000233721948e-7))
		tmp = single(0.5);
	else
		tmp = -s / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 48.1%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 59.0%

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

      1. mul-1-neg 59.0%

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

      2. unsub-neg 59.0%

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

   4. Simplified 59.0%

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

   5. Taylor expanded in x around inf 51.3%

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

      1. associate-*r/ 51.3%

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

      2. neg-mul-1 51.3%

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

   7. Simplified 51.3%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-s}{x}\\ \end{array} \]

Alternative 14: 34.5% accurate, 108.0× 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. Taylor expanded in x around 0 37.1%

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

3. Final simplification 37.1%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023194 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))