Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 7.3s

Alternatives: 12

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\left(e^{\sqrt[3]{\frac{x \cdot \frac{x}{s}}{s}}}\right)}^{\left(-\sqrt[3]{\frac{x}{s}}\right)}} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.9%

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

   1. distribute-frac-neg 99.9%

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

   2. exp-neg 99.9%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 53.1%

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

   5. sqrt-unprod 61.7%

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

   6. sqr-neg 61.7%

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

   7. sqrt-unprod 11.4%

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

   8. add-sqr-sqrt 26.8%

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

   9. div-inv 26.8%

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

   10. add-cube-cbrt 26.8%

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

   11. exp-prod 26.8%

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

   12. pow-flip 26.8%

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

3. Applied egg-rr 99.9%

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

   1. metadata-eval 99.9%

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

   2. pow-prod-up 99.9%

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

   3. inv-pow 99.9%

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

   4. clear-num 99.9%

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

   5. inv-pow 99.9%

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

   6. clear-num 99.9%

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

   7. associate-*r/ 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 3: 65.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 4:\\ \;\;\;\;0.5 + \left(\frac{x}{s} \cdot 0.25 + -0.020833333333333332 \cdot \frac{x \cdot \frac{x}{s}}{s \cdot \frac{s}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{x \cdot \frac{x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -5.0)
     0.5
     (if (<= t_0 4.0)
       (+
        0.5
        (+
         (* (/ x s) 0.25)
         (* -0.020833333333333332 (/ (* x (/ x s)) (* s (/ s x))))))
       (* 2.0 (/ 1.0 (* x (/ x (* s s)))))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 4.0f) {
		tmp = 0.5f + (((x / s) * 0.25f) + (-0.020833333333333332f * ((x * (x / s)) / (s * (s / x)))));
	} else {
		tmp = 2.0f * (1.0f / (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) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-5.0e0)) then
        tmp = 0.5e0
    else if (t_0 <= 4.0e0) then
        tmp = 0.5e0 + (((x / s) * 0.25e0) + ((-0.020833333333333332e0) * ((x * (x / s)) / (s * (s / x)))))
    else
        tmp = 2.0e0 * (1.0e0 / (x * (x / (s * s))))
    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(-5.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(4.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(Float32(x / s) * Float32(0.25)) + Float32(Float32(-0.020833333333333332) * Float32(Float32(x * Float32(x / s)) / Float32(s * Float32(s / x))))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(1.0) / Float32(x * Float32(x / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-5.0))
		tmp = single(0.5);
	elseif (t_0 <= single(4.0))
		tmp = single(0.5) + (((x / s) * single(0.25)) + (single(-0.020833333333333332) * ((x * (x / s)) / (s * (s / x)))));
	else
		tmp = single(2.0) * (single(1.0) / (x * (x / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.1%

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

   if -5 < (/.f32 (neg.f32 x) s) < 4

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 63.4%

      \[\leadsto \color{blue}{0.5 + \left(0.25 \cdot \frac{x}{s} + -0.020833333333333332 \cdot \frac{{x}^{3}}{{s}^{3}}\right)} \]
   3. Step-by-step derivation

      1. unpow3 63.4%

         \[\leadsto 0.5 + \left(0.25 \cdot \frac{x}{s} + -0.020833333333333332 \cdot \frac{\color{blue}{\left(x \cdot x\right) \cdot x}}{{s}^{3}}\right) \]

      2. cube-mult 63.4%

         \[\leadsto 0.5 + \left(0.25 \cdot \frac{x}{s} + -0.020833333333333332 \cdot \frac{\left(x \cdot x\right) \cdot x}{\color{blue}{s \cdot \left(s \cdot s\right)}}\right) \]

      3. times-frac 82.8%

         \[\leadsto 0.5 + \left(0.25 \cdot \frac{x}{s} + -0.020833333333333332 \cdot \color{blue}{\left(\frac{x \cdot x}{s} \cdot \frac{x}{s \cdot s}\right)}\right) \]

   4. Applied egg-rr 82.8%

      \[\leadsto 0.5 + \left(0.25 \cdot \frac{x}{s} + -0.020833333333333332 \cdot \color{blue}{\left(\frac{x \cdot x}{s} \cdot \frac{x}{s \cdot s}\right)}\right) \]
   5. Step-by-step derivation

      1. associate-/l* 83.1%

         \[\leadsto 0.5 + \left(0.25 \cdot \frac{x}{s} + -0.020833333333333332 \cdot \left(\color{blue}{\frac{x}{\frac{s}{x}}} \cdot \frac{x}{s \cdot s}\right)\right) \]

      2. associate-/r* 96.1%

         \[\leadsto 0.5 + \left(0.25 \cdot \frac{x}{s} + -0.020833333333333332 \cdot \left(\frac{x}{\frac{s}{x}} \cdot \color{blue}{\frac{\frac{x}{s}}{s}}\right)\right) \]

      3. frac-times 96.1%

         \[\leadsto 0.5 + \left(0.25 \cdot \frac{x}{s} + -0.020833333333333332 \cdot \color{blue}{\frac{x \cdot \frac{x}{s}}{\frac{s}{x} \cdot s}}\right) \]

   6. Applied egg-rr 96.1%

      \[\leadsto 0.5 + \left(0.25 \cdot \frac{x}{s} + -0.020833333333333332 \cdot \color{blue}{\frac{x \cdot \frac{x}{s}}{\frac{s}{x} \cdot s}}\right) \]

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 77.4%

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

      1. mul-1-neg 77.4%

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

      2. unsub-neg 77.4%

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

      3. unpow2 77.4%

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

      4. unpow2 77.4%

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

      5. times-frac 71.0%

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

   4. Simplified 71.0%

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

   5. Taylor expanded in x around inf 77.2%

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

      1. unpow2 77.2%

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

      2. unpow2 77.2%

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

   7. Simplified 77.2%

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

      1. associate-/r* 80.4%

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

      2. div-inv 80.4%

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

   9. Applied egg-rr 80.4%

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

       1. clear-num 80.4%

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

       2. frac-times 80.7%

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

       3. metadata-eval 80.7%

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

   11. Applied egg-rr 80.7%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 4:\\ \;\;\;\;0.5 + \left(\frac{x}{s} \cdot 0.25 + -0.020833333333333332 \cdot \frac{x \cdot \frac{x}{s}}{s \cdot \frac{s}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{x \cdot \frac{x}{s \cdot s}}\\ \end{array} \]

Alternative 4: 62.7% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 0.20000000298023224e0) then
        tmp = 0.5e0
    else
        tmp = 2.0e0 * (((s * s) / x) * (1.0e0 / x))
    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(2.0) * Float32(Float32(Float32(s * s) / x) * Float32(Float32(1.0) / x)));
	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(2.0) * (((s * s) / x) * (single(1.0) / x));
	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. Taylor expanded in x around 0 51.5%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 76.9%

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

      1. mul-1-neg 76.9%

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

      2. unsub-neg 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. times-frac 70.6%

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

   4. Simplified 70.6%

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

   5. Taylor expanded in x around inf 76.6%

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

      1. unpow2 76.6%

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

      2. unpow2 76.6%

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

   7. Simplified 76.6%

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

      1. associate-/r* 79.8%

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

      2. div-inv 79.8%

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

   9. Applied egg-rr 79.8%

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

4. Final simplification 61.8%

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

Alternative 5: 63.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{x \cdot \frac{x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.20000000298023224f) {
		tmp = 0.5f;
	} else {
		tmp = 2.0f * (1.0f / (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) <= 0.20000000298023224e0) then
        tmp = 0.5e0
    else
        tmp = 2.0e0 * (1.0e0 / (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(0.20000000298023224))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(1.0) / 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(0.20000000298023224))
		tmp = single(0.5);
	else
		tmp = single(2.0) * (single(1.0) / (x * (x / (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. Taylor expanded in x around 0 51.5%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 76.9%

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

      1. mul-1-neg 76.9%

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

      2. unsub-neg 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. times-frac 70.6%

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

   4. Simplified 70.6%

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

   5. Taylor expanded in x around inf 76.6%

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

      1. unpow2 76.6%

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

      2. unpow2 76.6%

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

   7. Simplified 76.6%

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

      1. associate-/r* 79.8%

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

      2. div-inv 79.8%

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

   9. Applied egg-rr 79.8%

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

       1. clear-num 79.8%

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

       2. frac-times 80.1%

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

       3. metadata-eval 80.1%

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

   11. Applied egg-rr 80.1%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{x \cdot \frac{x}{s \cdot s}}\\ \end{array} \]

Alternative 6: 58.5% accurate, 7.6× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.20000000298023224f) {
		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) <= 0.20000000298023224e0) 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(0.20000000298023224))
		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(0.20000000298023224))
		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) < 0.200000003

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 51.5%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 76.9%

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

      1. mul-1-neg 76.9%

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

      2. unsub-neg 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. times-frac 70.6%

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

   4. Simplified 70.6%

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

   5. Taylor expanded in x around inf 76.6%

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

      1. unpow2 76.6%

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

      2. unpow2 76.6%

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

   7. Simplified 76.6%

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

      1. times-frac 70.3%

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

   9. Applied egg-rr 70.3%

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

4. Final simplification 58.3%

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

Alternative 7: 61.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 20000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 20000.0) 0.5 (* 2.0 (/ (* s s) (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 20000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 2.0f * ((s * s) / (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 20000.0e0) then
        tmp = 0.5e0
    else
        tmp = 2.0e0 * ((s * s) / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(20000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(s * s) / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(20000.0))
		tmp = single(0.5);
	else
		tmp = single(2.0) * ((s * s) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 2e4

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 50.3%

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

   if 2e4 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 80.8%

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

      1. mul-1-neg 80.8%

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

      2. unsub-neg 80.8%

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

      3. unpow2 80.8%

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

      4. unpow2 80.8%

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

      5. times-frac 73.9%

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

   4. Simplified 73.9%

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

   5. Taylor expanded in x around inf 80.6%

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

      1. unpow2 80.6%

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

      2. unpow2 80.6%

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

   7. Simplified 80.6%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 20000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \]

Alternative 8: 62.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{\frac{x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.20000000298023224f) {
		tmp = 0.5f;
	} else {
		tmp = (2.0f / 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) <= 0.20000000298023224e0) then
        tmp = 0.5e0
    else
        tmp = (2.0e0 / 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(0.20000000298023224))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(2.0) / 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(0.20000000298023224))
		tmp = single(0.5);
	else
		tmp = (single(2.0) / x) / (x / (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. Taylor expanded in x around 0 51.5%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 76.9%

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

      1. mul-1-neg 76.9%

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

      2. unsub-neg 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. times-frac 70.6%

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

   4. Simplified 70.6%

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

   5. Taylor expanded in x around inf 76.6%

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

      1. unpow2 76.6%

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

      2. unpow2 76.6%

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

   7. Simplified 76.6%

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

      1. associate-/r* 79.8%

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

      2. div-inv 79.8%

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

   9. Applied egg-rr 79.8%

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

       1. *-commutative 79.8%

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

       2. associate-*r* 79.8%

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

       3. clear-num 79.8%

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

       4. un-div-inv 79.8%

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

       5. un-div-inv 79.8%

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

   11. Applied egg-rr 79.8%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{\frac{x}{s \cdot s}}\\ \end{array} \]

Alternative 9: 49.4% 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 99.9%

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

   2. Taylor expanded in x around 0 28.2%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 63.4%

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

      1. mul-1-neg 63.4%

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

      2. unsub-neg 63.4%

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

   4. Simplified 63.4%

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

4. Final simplification 50.1%

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

Alternative 10: 47.9% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s))) (if (<= t_0 0.20000000298023224) 0.5 (/ 1.0 t_0))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 0.20000000298023224f) {
		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 <= 0.20000000298023224e0) 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(0.20000000298023224))
		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(0.20000000298023224))
		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) < 0.200000003

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 51.5%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 42.1%

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

      1. mul-1-neg 42.1%

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

      2. unsub-neg 42.1%

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

   4. Simplified 42.1%

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

   5. Taylor expanded in x around inf 42.1%

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

      1. neg-mul-1 42.1%

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

      2. distribute-neg-frac 42.1%

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

   7. Simplified 42.1%

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

4. Final simplification 48.1%

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

Alternative 11: 46.2% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000116860974e-7) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000116860974e-7f) {
		tmp = s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.0000000116860974e-7)) then
        tmp = s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000116860974e-7))
		tmp = Float32(s / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.0000000116860974e-7))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000001e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 49.1%

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

      1. mul-1-neg 49.1%

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

      2. unsub-neg 49.1%

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

   4. Simplified 49.1%

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

   5. Taylor expanded in x around inf 46.8%

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

      1. associate-*r/ 46.8%

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

      2. neg-mul-1 46.8%

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

   7. Simplified 46.8%

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

      1. expm1-log1p-u 46.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-s}{x}\right)\right)} \]

      2. expm1-udef 96.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-s}{x}\right)} - 1} \]

      3. frac-2neg 96.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-\left(-s\right)}{-x}}\right)} - 1 \]

      4. remove-double-neg 96.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{s}}{-x}\right)} - 1 \]

      5. add-sqr-sqrt 96.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{s}{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}\right)} - 1 \]

      6. sqrt-unprod 96.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{s}{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}\right)} - 1 \]

      7. sqr-neg 96.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{s}{\sqrt{\color{blue}{x \cdot x}}}\right)} - 1 \]

      8. sqrt-prod -0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{s}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)} - 1 \]

      9. add-sqr-sqrt 96.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{s}{\color{blue}{x}}\right)} - 1 \]

   9. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{s}{x}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 46.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{s}{x}\right)\right)} \]

       2. expm1-log1p 46.8%

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

   11. Simplified 46.8%

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

   if -1.00000001e-7 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 47.6%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 12: 34.8% 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 99.9%

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

2. Taylor expanded in x around 0 35.2%

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

3. Final simplification 35.2%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023255 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))