Logistic distribution

Percentage Accurate: 99.5% → 99.2%

Time: 10.3s

Alternatives: 13

Speedup: 1.5×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t_0\\ \frac{t_0}{\left(s \cdot t_1\right) \cdot t_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t_0\\ \frac{t_0}{\left(s \cdot t_1\right) \cdot t_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Alternative 1: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ \frac{\frac{1}{s}}{{\left(e^{{\left(\sqrt[3]{t_0}\right)}^{2}}\right)}^{\left({t_0}^{0.3333333333333333}\right)} + \left(2 + e^{\frac{\left|x\right|}{-s}}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) s)))
   (/
    (/ 1.0 s)
    (+
     (pow (exp (pow (cbrt t_0) 2.0)) (pow t_0 0.3333333333333333))
     (+ 2.0 (exp (/ (fabs x) (- s))))))))

C

float code(float x, float s) {
	float t_0 = fabsf(x) / s;
	return (1.0f / s) / (powf(expf(powf(cbrtf(t_0), 2.0f)), powf(t_0, 0.3333333333333333f)) + (2.0f + expf((fabsf(x) / -s))));
}

Julia

function code(x, s)
	t_0 = Float32(abs(x) / s)
	return Float32(Float32(Float32(1.0) / s) / Float32((exp((cbrt(t_0) ^ Float32(2.0))) ^ (t_0 ^ Float32(0.3333333333333333))) + Float32(Float32(2.0) + exp(Float32(abs(x) / Float32(-s))))))
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 99.8%

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

   2. exp-prod 99.8%

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

   3. pow2 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in x around 0 99.8%

   \[\leadsto \frac{\frac{1}{s}}{{\left(e^{{\left(\sqrt[3]{\frac{\left|x\right|}{s}}\right)}^{2}}\right)}^{\color{blue}{\left({\left(\frac{\left|x\right|}{s}\right)}^{0.3333333333333333}\right)}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]

7. Final simplification 99.8%

   \[\leadsto \frac{\frac{1}{s}}{{\left(e^{{\left(\sqrt[3]{\frac{\left|x\right|}{s}}\right)}^{2}}\right)}^{\left({\left(\frac{\left|x\right|}{s}\right)}^{0.3333333333333333}\right)} + \left(2 + e^{\frac{\left|x\right|}{-s}}\right)} \]

Alternative 2: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + \left(2 + e^{\frac{\left|x\right|}{s}}\right)\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (* s (+ (exp (/ (- (fabs x)) s)) (+ 2.0 (exp (/ (fabs x) s)))))))

C

float code(float x, float s) {
	return 1.0f / (s * (expf((-fabsf(x) / s)) + (2.0f + expf((fabsf(x) / s)))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * (exp((-abs(x) / s)) + (2.0e0 + exp((abs(x) / s)))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32(exp(Float32(Float32(-abs(x)) / s)) + Float32(Float32(2.0) + exp(Float32(abs(x) / s))))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (s * (exp((-abs(x) / s)) + (single(2.0) + exp((abs(x) / s)))));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around 0 99.8%

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

   1. mul-1-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 86.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{s} \cdot \frac{x}{s}\\ \mathbf{if}\;\left|x\right| \leq 5.999999759184749 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{t_0 \cdot \frac{x}{\frac{s \cdot s}{x}} - 16}{t_0 - 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{\sqrt[3]{\left(s \cdot s\right) \cdot \left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (* (/ x s) (/ x s))))
   (if (<= (fabs x) 5.999999759184749e-13)
     (/ (/ 1.0 s) (/ (- (* t_0 (/ x (/ (* s s) x))) 16.0) (- t_0 4.0)))
     (/
      (/ 1.0 s)
      (+ 4.0 (/ (* x x) (cbrt (* (* s s) (* (* s s) (* s s))))))))))

C

float code(float x, float s) {
	float t_0 = (x / s) * (x / s);
	float tmp;
	if (fabsf(x) <= 5.999999759184749e-13f) {
		tmp = (1.0f / s) / (((t_0 * (x / ((s * s) / x))) - 16.0f) / (t_0 - 4.0f));
	} else {
		tmp = (1.0f / s) / (4.0f + ((x * x) / cbrtf(((s * s) * ((s * s) * (s * s))))));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(x / s) * Float32(x / s))
	tmp = Float32(0.0)
	if (abs(x) <= Float32(5.999999759184749e-13))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(Float32(t_0 * Float32(x / Float32(Float32(s * s) / x))) - Float32(16.0)) / Float32(t_0 - Float32(4.0))));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x * x) / cbrt(Float32(Float32(s * s) * Float32(Float32(s * s) * Float32(s * s)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 5.99999976e-13

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in s around inf 70.0%

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

      1. associate-+r+ 70.0%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 70.0%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 70.0%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 70.0%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 70.0%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 70.0%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 70.0%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 70.0%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 70.0%

         \[\leadsto \frac{\frac{1}{s}}{\frac{x \cdot x}{\color{blue}{s \cdot s}} + 4} \]

   6. Simplified 70.0%

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

      1. flip-+ 52.0%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{\frac{x \cdot x}{s \cdot s} \cdot \frac{x \cdot x}{s \cdot s} - 4 \cdot 4}{\frac{x \cdot x}{s \cdot s} - 4}}} \]

      2. times-frac 52.0%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} \cdot \frac{x \cdot x}{s \cdot s} - 4 \cdot 4}{\frac{x \cdot x}{s \cdot s} - 4}} \]

      3. times-frac 52.1%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} - 4 \cdot 4}{\frac{x \cdot x}{s \cdot s} - 4}} \]

      4. metadata-eval 52.1%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \color{blue}{16}}{\frac{x \cdot x}{s \cdot s} - 4}} \]

      5. times-frac 82.2%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - 16}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}} - 4}} \]

   8. Applied egg-rr 82.2%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - 16}{\frac{x}{s} \cdot \frac{x}{s} - 4}}} \]
   9. Step-by-step derivation

      1. frac-times 69.1%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\frac{x \cdot x}{s \cdot s}} \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - 16}{\frac{x}{s} \cdot \frac{x}{s} - 4}} \]

      2. add-cbrt-cube 37.5%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\frac{x \cdot x}{\color{blue}{\sqrt[3]{\left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right) \cdot \left(s \cdot s\right)}}} \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - 16}{\frac{x}{s} \cdot \frac{x}{s} - 4}} \]

      3. associate-/l* 52.1%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\frac{x}{\frac{\sqrt[3]{\left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right) \cdot \left(s \cdot s\right)}}{x}}} \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - 16}{\frac{x}{s} \cdot \frac{x}{s} - 4}} \]

      4. add-cbrt-cube 82.7%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\frac{x}{\frac{\color{blue}{s \cdot s}}{x}} \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - 16}{\frac{x}{s} \cdot \frac{x}{s} - 4}} \]

   10. Applied egg-rr 82.7%

       \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\frac{x}{\frac{s \cdot s}{x}}} \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - 16}{\frac{x}{s} \cdot \frac{x}{s} - 4}} \]

   if 5.99999976e-13 < (fabs.f32 x)

   1. Initial program 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 41.8%

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

      1. associate-+r+ 41.8%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 41.8%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 41.8%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 82.8%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 82.8%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 82.8%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 82.8%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 82.8%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 82.8%

         \[\leadsto \frac{\frac{1}{s}}{\frac{x \cdot x}{\color{blue}{s \cdot s}} + 4} \]

   6. Simplified 82.8%

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

      1. add-cbrt-cube 93.5%

         \[\leadsto \frac{\frac{1}{s}}{\frac{x \cdot x}{\color{blue}{\sqrt[3]{\left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right) \cdot \left(s \cdot s\right)}}} + 4} \]

   8. Applied egg-rr 93.5%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5.999999759184749 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \frac{x}{\frac{s \cdot s}{x}} - 16}{\frac{x}{s} \cdot \frac{x}{s} - 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{\sqrt[3]{\left(s \cdot s\right) \cdot \left(\left(s \cdot s\right) \cdot \left(s \cdot s\right)\right)}}}\\ \end{array} \]

Alternative 4: 96.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{s}}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{s} \cdot \frac{x}{s}\right)\right) + 4} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (/ 1.0 s) (+ (log1p (expm1 (* (/ x s) (/ x s)))) 4.0)))

C

float code(float x, float s) {
	return (1.0f / s) / (log1pf(expm1f(((x / s) * (x / s)))) + 4.0f);
}

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / Float32(log1p(expm1(Float32(Float32(x / s) * Float32(x / s)))) + Float32(4.0)))
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 52.8%

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

   1. associate-+r+ 52.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

   2. distribute-lft1-in 52.8%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

   3. metadata-eval 52.8%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

   4. mul0-lft 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

   5. associate-+r+ 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

   6. metadata-eval 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

   7. unpow2 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

   8. sqr-abs 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

   9. unpow2 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\frac{x \cdot x}{\color{blue}{s \cdot s}} + 4} \]

6. Simplified 77.8%

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

   1. log1p-expm1-u 89.0%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x \cdot x}{s \cdot s}\right)\right)} + 4} \]

   2. times-frac 98.1%

      \[\leadsto \frac{\frac{1}{s}}{\mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{x}{s} \cdot \frac{x}{s}}\right)\right) + 4} \]

8. Applied egg-rr 98.1%

   \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{s} \cdot \frac{x}{s}\right)\right)} + 4} \]

9. Final simplification 98.1%

   \[\leadsto \frac{\frac{1}{s}}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{s} \cdot \frac{x}{s}\right)\right) + 4} \]

Alternative 5: 84.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{s} \cdot \frac{x}{s}\\ \mathbf{if}\;\left|x\right| \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{t_0 \cdot t_0 - 16}{t_0 - 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(x \cdot x\right) \cdot \frac{1}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (* (/ x s) (/ x s))))
   (if (<= (fabs x) 1.9999999996399175e-23)
     (/ (/ 1.0 s) (/ (- (* t_0 t_0) 16.0) (- t_0 4.0)))
     (/ (/ 1.0 s) (+ 4.0 (* (* x x) (/ 1.0 (* s s))))))))

C

float code(float x, float s) {
	float t_0 = (x / s) * (x / s);
	float tmp;
	if (fabsf(x) <= 1.9999999996399175e-23f) {
		tmp = (1.0f / s) / (((t_0 * t_0) - 16.0f) / (t_0 - 4.0f));
	} else {
		tmp = (1.0f / s) / (4.0f + ((x * x) * (1.0f / (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) * (x / s)
    if (abs(x) <= 1.9999999996399175e-23) then
        tmp = (1.0e0 / s) / (((t_0 * t_0) - 16.0e0) / (t_0 - 4.0e0))
    else
        tmp = (1.0e0 / s) / (4.0e0 + ((x * x) * (1.0e0 / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(x / s) * Float32(x / s))
	tmp = Float32(0.0)
	if (abs(x) <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(Float32(t_0 * t_0) - Float32(16.0)) / Float32(t_0 - Float32(4.0))));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x * x) * Float32(Float32(1.0) / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = (x / s) * (x / s);
	tmp = single(0.0);
	if (abs(x) <= single(1.9999999996399175e-23))
		tmp = (single(1.0) / s) / (((t_0 * t_0) - single(16.0)) / (t_0 - single(4.0)));
	else
		tmp = (single(1.0) / s) / (single(4.0) + ((x * x) * (single(1.0) / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 2e-23

   1. Initial program 99.1%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in s around inf 54.7%

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

      1. associate-+r+ 54.7%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 54.7%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 54.7%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 54.7%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 54.7%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 54.7%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 54.7%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 54.7%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 54.7%

         \[\leadsto \frac{\frac{1}{s}}{\frac{x \cdot x}{\color{blue}{s \cdot s}} + 4} \]

   6. Simplified 54.7%

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

      1. flip-+ 54.7%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{\frac{x \cdot x}{s \cdot s} \cdot \frac{x \cdot x}{s \cdot s} - 4 \cdot 4}{\frac{x \cdot x}{s \cdot s} - 4}}} \]

      2. times-frac 54.7%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} \cdot \frac{x \cdot x}{s \cdot s} - 4 \cdot 4}{\frac{x \cdot x}{s \cdot s} - 4}} \]

      3. times-frac 54.7%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} - 4 \cdot 4}{\frac{x \cdot x}{s \cdot s} - 4}} \]

      4. metadata-eval 54.7%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \color{blue}{16}}{\frac{x \cdot x}{s \cdot s} - 4}} \]

      5. times-frac 79.9%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - 16}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}} - 4}} \]

   8. Applied egg-rr 79.9%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - 16}{\frac{x}{s} \cdot \frac{x}{s} - 4}}} \]

   if 2e-23 < (fabs.f32 x)

   1. Initial program 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 52.2%

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

      1. associate-+r+ 52.2%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 52.2%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 52.2%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 84.9%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 84.9%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 84.9%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 84.9%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 84.9%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 84.9%

         \[\leadsto \frac{\frac{1}{s}}{\frac{x \cdot x}{\color{blue}{s \cdot s}} + 4} \]

   6. Simplified 84.9%

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

      1. div-inv 86.0%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(x \cdot x\right) \cdot \frac{1}{s \cdot s}} + 4} \]

   8. Applied egg-rr 86.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - 16}{\frac{x}{s} \cdot \frac{x}{s} - 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(x \cdot x\right) \cdot \frac{1}{s \cdot s}}\\ \end{array} \]

Alternative 6: 91.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{s} \cdot \frac{x}{s}\\ \frac{\frac{1}{s}}{4 + \sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)}} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (* (/ x s) (/ x s))))
   (/ (/ 1.0 s) (+ 4.0 (cbrt (* t_0 (* t_0 t_0)))))))

C

float code(float x, float s) {
	float t_0 = (x / s) * (x / s);
	return (1.0f / s) / (4.0f + cbrtf((t_0 * (t_0 * t_0))));
}

Julia

function code(x, s)
	t_0 = Float32(Float32(x / s) * Float32(x / s))
	return Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + cbrt(Float32(t_0 * Float32(t_0 * t_0)))))
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 52.8%

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

   1. associate-+r+ 52.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

   2. distribute-lft1-in 52.8%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

   3. metadata-eval 52.8%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

   4. mul0-lft 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

   5. associate-+r+ 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

   6. metadata-eval 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

   7. unpow2 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

   8. sqr-abs 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

   9. unpow2 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\frac{x \cdot x}{\color{blue}{s \cdot s}} + 4} \]

6. Simplified 77.8%

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

   1. add-cbrt-cube 83.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\sqrt[3]{\left(\frac{x \cdot x}{s \cdot s} \cdot \frac{x \cdot x}{s \cdot s}\right) \cdot \frac{x \cdot x}{s \cdot s}}} + 4} \]

   2. times-frac 83.8%

      \[\leadsto \frac{\frac{1}{s}}{\sqrt[3]{\left(\color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} \cdot \frac{x \cdot x}{s \cdot s}\right) \cdot \frac{x \cdot x}{s \cdot s}} + 4} \]

   3. times-frac 83.8%

      \[\leadsto \frac{\frac{1}{s}}{\sqrt[3]{\left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)}\right) \cdot \frac{x \cdot x}{s \cdot s}} + 4} \]

   4. times-frac 89.7%

      \[\leadsto \frac{\frac{1}{s}}{\sqrt[3]{\left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)\right) \cdot \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)}} + 4} \]

8. Applied egg-rr 89.7%

   \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\sqrt[3]{\left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}} + 4} \]

9. Final simplification 89.7%

   \[\leadsto \frac{\frac{1}{s}}{4 + \sqrt[3]{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)\right)}} \]

Alternative 7: 79.5% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s} \cdot \frac{x}{s} + 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(x \cdot x\right) \cdot \frac{1}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999996399175e-23)
   (/ (/ 1.0 s) (+ (* (/ x s) (/ x s)) 4.0))
   (/ (/ 1.0 s) (+ 4.0 (* (* x x) (/ 1.0 (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999996399175e-23f) {
		tmp = (1.0f / s) / (((x / s) * (x / s)) + 4.0f);
	} else {
		tmp = (1.0f / s) / (4.0f + ((x * x) * (1.0f / (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 <= 1.9999999996399175e-23) then
        tmp = (1.0e0 / s) / (((x / s) * (x / s)) + 4.0e0)
    else
        tmp = (1.0e0 / s) / (4.0e0 + ((x * x) * (1.0e0 / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(Float32(x / s) * Float32(x / s)) + Float32(4.0)));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x * x) * Float32(Float32(1.0) / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999996399175e-23))
		tmp = (single(1.0) / s) / (((x / s) * (x / s)) + single(4.0));
	else
		tmp = (single(1.0) / s) / (single(4.0) + ((x * x) * (single(1.0) / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2e-23

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 56.6%

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

      1. associate-+r+ 56.6%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 56.6%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 56.6%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 72.2%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 72.2%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 72.2%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 72.2%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 72.2%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 72.2%

         \[\leadsto \frac{\frac{1}{s}}{\frac{x \cdot x}{\color{blue}{s \cdot s}} + 4} \]

   6. Simplified 72.2%

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

      1. times-frac 72.9%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}} + 4} \]

   8. Applied egg-rr 72.9%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}} + 4} \]

   if 2e-23 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in s around inf 47.7%

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

      1. associate-+r+ 47.7%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 47.7%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 47.7%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 85.4%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 85.4%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 85.4%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 85.4%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 85.4%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 85.4%

         \[\leadsto \frac{\frac{1}{s}}{\frac{x \cdot x}{\color{blue}{s \cdot s}} + 4} \]

   6. Simplified 85.4%

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

      1. div-inv 86.5%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(x \cdot x\right) \cdot \frac{1}{s \cdot s}} + 4} \]

   8. Applied egg-rr 86.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s} \cdot \frac{x}{s} + 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(x \cdot x\right) \cdot \frac{1}{s \cdot s}}\\ \end{array} \]

Alternative 8: 78.7% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s} \cdot \frac{x}{s} + 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999996399175e-23)
   (/ (/ 1.0 s) (+ (* (/ x s) (/ x s)) 4.0))
   (/ (/ 1.0 s) (+ 4.0 (/ (* x x) (* s s))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999996399175e-23f) {
		tmp = (1.0f / s) / (((x / s) * (x / s)) + 4.0f);
	} else {
		tmp = (1.0f / s) / (4.0f + ((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 <= 1.9999999996399175e-23) then
        tmp = (1.0e0 / s) / (((x / s) * (x / s)) + 4.0e0)
    else
        tmp = (1.0e0 / s) / (4.0e0 + ((x * x) / (s * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(Float32(x / s) * Float32(x / s)) + Float32(4.0)));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x * x) / Float32(s * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999996399175e-23))
		tmp = (single(1.0) / s) / (((x / s) * (x / s)) + single(4.0));
	else
		tmp = (single(1.0) / s) / (single(4.0) + ((x * x) / (s * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2e-23

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 56.6%

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

      1. associate-+r+ 56.6%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 56.6%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 56.6%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 72.2%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 72.2%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 72.2%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 72.2%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 72.2%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 72.2%

         \[\leadsto \frac{\frac{1}{s}}{\frac{x \cdot x}{\color{blue}{s \cdot s}} + 4} \]

   6. Simplified 72.2%

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

      1. times-frac 72.9%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}} + 4} \]

   8. Applied egg-rr 72.9%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}} + 4} \]

   if 2e-23 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in s around inf 47.7%

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

      1. associate-+r+ 47.7%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 47.7%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 47.7%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 85.4%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 85.4%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 85.4%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 85.4%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 85.4%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 85.4%

         \[\leadsto \frac{\frac{1}{s}}{\frac{x \cdot x}{\color{blue}{s \cdot s}} + 4} \]

   6. Simplified 85.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s} \cdot \frac{x}{s} + 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 9: 76.2% accurate, 47.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{s}}{\frac{x}{s} \cdot \frac{x}{s} + 4} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (/ 1.0 s) (+ (* (/ x s) (/ x s)) 4.0)))

C

float code(float x, float s) {
	return (1.0f / s) / (((x / s) * (x / s)) + 4.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / s) / (((x / s) * (x / s)) + 4.0e0)
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / Float32(Float32(Float32(x / s) * Float32(x / s)) + Float32(4.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / s) / (((x / s) * (x / s)) + single(4.0));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 52.8%

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

   1. associate-+r+ 52.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

   2. distribute-lft1-in 52.8%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

   3. metadata-eval 52.8%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

   4. mul0-lft 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

   5. associate-+r+ 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

   6. metadata-eval 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

   7. unpow2 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

   8. sqr-abs 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

   9. unpow2 77.8%

      \[\leadsto \frac{\frac{1}{s}}{\frac{x \cdot x}{\color{blue}{s \cdot s}} + 4} \]

6. Simplified 77.8%

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

   1. times-frac 74.2%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}} + 4} \]

8. Applied egg-rr 74.2%

   \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}} + 4} \]

9. Final simplification 74.2%

   \[\leadsto \frac{\frac{1}{s}}{\frac{x}{s} \cdot \frac{x}{s} + 4} \]

Alternative 10: 45.6% accurate, 67.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 3.999999989900971e-6) (/ 0.25 s) (/ 1.0 (* x (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 3.999999989900971e-6f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (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 <= 3.999999989900971e-6) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (x * (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(3.999999989900971e-6))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(3.999999989900971e-6))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (x * (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999e-6

   1. Initial program 99.6%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.6%

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

      2. associate-*r/ 99.6%

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

      3. associate-/l* 99.6%

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

      4. distribute-frac-neg 99.6%

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

      5. exp-neg 99.6%

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

      6. associate-/r/ 99.6%

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

      7. /-rgt-identity 99.6%

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

      8. associate-*l* 99.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 39.1%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]

   if 3.99999999e-6 < x

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 2.6%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]

   6. Simplified 2.6%

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

      1. add-exp-log 2.6%

         \[\leadsto \frac{1}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \color{blue}{e^{\log \left(\frac{x \cdot x}{s}\right)}}} \]

      2. associate-/l* 2.6%

         \[\leadsto \frac{1}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - e^{\log \color{blue}{\left(\frac{x}{\frac{s}{x}}\right)}}} \]

   8. Applied egg-rr 2.6%

      \[\leadsto \frac{1}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \color{blue}{e^{\log \left(\frac{x}{\frac{s}{x}}\right)}}} \]

   9. Taylor expanded in x around inf 78.6%

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

       1. unpow2 78.6%

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

       2. associate-*r/ 78.6%

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

   11. Simplified 78.6%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 11: 45.6% accurate, 67.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 3.999999989900971e-6) (/ 0.25 s) (/ 1.0 (/ (* x x) s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 3.999999989900971e-6f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / ((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 <= 3.999999989900971e-6) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / ((x * x) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(3.999999989900971e-6))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * x) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(3.999999989900971e-6))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / ((x * x) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999e-6

   1. Initial program 99.6%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.6%

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

      2. associate-*r/ 99.6%

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

      3. associate-/l* 99.6%

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

      4. distribute-frac-neg 99.6%

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

      5. exp-neg 99.6%

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

      6. associate-/r/ 99.6%

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

      7. /-rgt-identity 99.6%

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

      8. associate-*l* 99.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 39.1%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]

   if 3.99999999e-6 < x

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 2.6%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]

   6. Simplified 2.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 78.6%

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

      1. unpow2 78.6%

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

   9. Simplified 78.6%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \end{array} \]

Alternative 12: 44.8% accurate, 87.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 3.999999989900971e-6) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 3.999999989900971e-6f) {
		tmp = 0.25f / s;
	} else {
		tmp = 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 <= 3.999999989900971e-6) then
        tmp = 0.25e0 / s
    else
        tmp = s / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(3.999999989900971e-6))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(3.999999989900971e-6))
		tmp = single(0.25) / s;
	else
		tmp = s / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999e-6

   1. Initial program 99.6%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.6%

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

      2. associate-*r/ 99.6%

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

      3. associate-/l* 99.6%

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

      4. distribute-frac-neg 99.6%

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

      5. exp-neg 99.6%

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

      6. associate-/r/ 99.6%

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

      7. /-rgt-identity 99.6%

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

      8. associate-*l* 99.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 39.1%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]

   if 3.99999999e-6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 33.1%

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

      1. associate-+r+ 33.1%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 33.1%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 33.1%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 87.7%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 87.7%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 87.7%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 87.7%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 87.7%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 87.7%

         \[\leadsto \frac{\frac{1}{s}}{\frac{x \cdot x}{\color{blue}{s \cdot s}} + 4} \]

   6. Simplified 87.7%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x \cdot x}{s \cdot s} + 4}} \]

   7. Taylor expanded in s around 0 76.1%

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

      1. unpow2 76.1%

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

   9. Simplified 76.1%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 13: 26.6% accurate, 206.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 s))

C

float code(float x, float s) {
	return 0.25f / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / s
end function

Julia

function code(x, s)
	return Float32(Float32(0.25) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.25) / s;
end

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Step-by-step derivation

   1. *-lft-identity 99.7%

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

   2. associate-*r/ 99.7%

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

   3. associate-/l* 99.7%

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

   4. distribute-frac-neg 99.7%

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

   5. exp-neg 99.7%

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

   6. associate-/r/ 99.7%

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

   7. /-rgt-identity 99.7%

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

   8. associate-*l* 99.7%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 28.9%

   \[\leadsto \color{blue}{\frac{0.25}{s}} \]

5. Final simplification 28.9%

   \[\leadsto \frac{0.25}{s} \]

Reproduce

herbie shell --seed 2023200 
(FPCore (x s)
  :name "Logistic distribution"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ (exp (/ (- (fabs x)) s)) (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))