Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 14.3s

Alternatives: 17

Speedup: 1.0×

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. distribute-lft-in 99.8%

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

   3. *-rgt-identity 99.8%

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

   4. fabs-neg 99.8%

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

   5. distribute-frac-neg 99.8%

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

   6. exp-neg 99.8%

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

   7. associate-*r/ 99.8%

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

   8. *-rgt-identity 99.8%

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

   9. *-lft-identity 99.8%

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

   10. metadata-eval 99.8%

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

   11. times-frac 99.8%

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

   12. neg-mul-1 99.8%

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

   13. neg-mul-1 99.8%

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

   14. fabs-neg 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 99.8%

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

   1. mul-1-neg 99.8%

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

   2. exp-neg 99.8%

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

   3. add-sqr-sqrt 50.3%

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

   4. fabs-sqr 50.3%

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

   5. add-sqr-sqrt 98.6%

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

6. Applied egg-rr 98.6%

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

   1. rec-exp 98.6%

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

   2. distribute-neg-frac 98.6%

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

8. Simplified 98.6%

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

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-unprod 98.6%

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

   3. sqr-neg 98.6%

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

   4. distribute-frac-neg 98.6%

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

   5. distribute-frac-neg 98.6%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 62.4%

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

   8. expm1-log1p-u 62.3%

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

   9. expm1-udef 62.3%

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

10. Applied egg-rr 99.7%

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

    1. expm1-def 99.7%

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

    2. expm1-log1p 99.8%

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

    3. +-commutative 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 3: 95.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / ((s * 2.0f) * (1.0f + 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 * 2.0e0) * (1.0e0 + exp((abs(x) / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 99.8%

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

   1. mul-1-neg 99.8%

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

   2. exp-neg 99.8%

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

   3. add-sqr-sqrt 50.3%

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

   4. fabs-sqr 50.3%

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

   5. add-sqr-sqrt 98.6%

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

6. Applied egg-rr 98.6%

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

   1. rec-exp 98.6%

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

   2. distribute-neg-frac 98.6%

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

8. Simplified 98.6%

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

9. Taylor expanded in x around 0 96.7%

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

    1. associate-*r* 96.7%

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

11. Simplified 96.7%

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

12. Final simplification 96.7%

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

Alternative 4: 60.7% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (* (* s 2.0) (+ 1.0 (pow E (/ x s))))))

C

float code(float x, float s) {
	return 1.0f / ((s * 2.0f) * (1.0f + powf(((float) M_E), (x / s))));
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(s * Float32(2.0)) * Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(x / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((s * single(2.0)) * (single(1.0) + (single(2.71828182845904523536) ^ (x / s))));
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 99.8%

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

   1. mul-1-neg 99.8%

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

   2. exp-neg 99.8%

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

   3. add-sqr-sqrt 50.3%

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

   4. fabs-sqr 50.3%

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

   5. add-sqr-sqrt 98.6%

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

6. Applied egg-rr 98.6%

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

   1. rec-exp 98.6%

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

   2. distribute-neg-frac 98.6%

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

8. Simplified 98.6%

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

9. Taylor expanded in x around 0 96.7%

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

    1. associate-*r* 96.7%

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

11. Simplified 96.7%

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

    1. *-un-lft-identity 96.7%

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

    2. exp-prod 96.7%

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

    3. add-sqr-sqrt 48.2%

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

    4. fabs-sqr 48.2%

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

    5. add-sqr-sqrt 59.6%

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

13. Applied egg-rr 59.6%

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

    1. exp-1-e 59.6%

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

15. Simplified 59.6%

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

16. Final simplification 59.6%

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

Alternative 5: 60.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{s \cdot 2}}{1 + e^{\frac{x}{s}}} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	return (1.0f / (s * 2.0f)) / (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 / (s * 2.0e0)) / (1.0e0 + exp((x / s)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 99.8%

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

   1. mul-1-neg 99.8%

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

   2. exp-neg 99.8%

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

   3. add-sqr-sqrt 50.3%

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

   4. fabs-sqr 50.3%

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

   5. add-sqr-sqrt 98.6%

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

6. Applied egg-rr 98.6%

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

   1. rec-exp 98.6%

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

   2. distribute-neg-frac 98.6%

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

8. Simplified 98.6%

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

9. Taylor expanded in x around 0 96.7%

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

    1. associate-*r* 96.7%

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

11. Simplified 96.7%

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

    1. expm1-log1p-u 95.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left(2 \cdot s\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}\right)\right)} \]

    2. expm1-udef 95.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\left(2 \cdot s\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}\right)} - 1} \]

    3. associate-/r* 95.3%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{2 \cdot s}}{1 + e^{\frac{\left|x\right|}{s}}}}\right)} - 1 \]

    4. *-commutative 95.3%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{\color{blue}{s \cdot 2}}}{1 + e^{\frac{\left|x\right|}{s}}}\right)} - 1 \]

    5. div-inv 95.3%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s \cdot 2}}{1 + e^{\color{blue}{\left|x\right| \cdot \frac{1}{s}}}}\right)} - 1 \]

    6. add-sqr-sqrt 47.5%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s \cdot 2}}{1 + e^{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{s}}}\right)} - 1 \]

    7. fabs-sqr 47.5%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s \cdot 2}}{1 + e^{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{s}}}\right)} - 1 \]

    8. add-sqr-sqrt 58.3%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s \cdot 2}}{1 + e^{\color{blue}{x} \cdot \frac{1}{s}}}\right)} - 1 \]

    9. div-inv 58.3%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s \cdot 2}}{1 + e^{\color{blue}{\frac{x}{s}}}}\right)} - 1 \]

13. Applied egg-rr 58.3%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{s \cdot 2}}{1 + e^{\frac{x}{s}}}\right)} - 1} \]
14. Step-by-step derivation

    1. expm1-def 58.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s \cdot 2}}{1 + e^{\frac{x}{s}}}\right)\right)} \]

    2. expm1-log1p 59.6%

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

15. Simplified 59.6%

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

16. Final simplification 59.6%

    \[\leadsto \frac{\frac{1}{s \cdot 2}}{1 + e^{\frac{x}{s}}} \]

Alternative 6: 46.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3 \cdot \frac{{x}^{2}}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.999999987845058e-8) (/ 0.25 s) (/ 1.0 (* 3.0 (/ (pow x 2.0) s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.999999987845058e-8f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (3.0f * (powf(x, 2.0f) / 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 <= 1.999999987845058e-8) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (3.0e0 * ((x ** 2.0e0) / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.999999987845058e-8))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(3.0) * Float32((x ^ Float32(2.0)) / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.999999987845058e-8))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (single(3.0) * ((x ^ single(2.0)) / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999e-8

   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. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 34.5%

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

   if 1.99999999e-8 < 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%

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

   5. Applied egg-rr 98.8%

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

      1. associate-+r+ 98.8%

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

      2. *-lft-identity 98.8%

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

      3. distribute-rgt-in 98.8%

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

      4. +-commutative 98.8%

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

      5. *-commutative 98.8%

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

      6. fma-udef 98.8%

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

      7. distribute-rgt-in 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-in 98.8%

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

      10. associate-*r* 98.8%

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

      11. distribute-lft-out 98.8%

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

      12. *-lft-identity 98.8%

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

      13. distribute-rgt-in 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 72.6%

      \[\leadsto \frac{1}{\color{blue}{3 \cdot \frac{{x}^{2}}{s} + \left(4 \cdot s + 4 \cdot x\right)}} \]

   9. Taylor expanded in x around inf 72.6%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3 \cdot \frac{{x}^{2}}{s}}\\ \end{array} \]

Alternative 7: 46.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{2} \cdot \frac{3}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.999999987845058e-8) (/ 0.25 s) (/ 1.0 (* (pow x 2.0) (/ 3.0 s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.999999987845058e-8f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (powf(x, 2.0f) * (3.0f / 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 <= 1.999999987845058e-8) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / ((x ** 2.0e0) * (3.0e0 / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.999999987845058e-8))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32((x ^ Float32(2.0)) * Float32(Float32(3.0) / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.999999987845058e-8))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / ((x ^ single(2.0)) * (single(3.0) / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999e-8

   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. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 34.5%

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

   if 1.99999999e-8 < 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%

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

   5. Applied egg-rr 98.8%

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

      1. associate-+r+ 98.8%

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

      2. *-lft-identity 98.8%

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

      3. distribute-rgt-in 98.8%

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

      4. +-commutative 98.8%

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

      5. *-commutative 98.8%

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

      6. fma-udef 98.8%

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

      7. distribute-rgt-in 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-in 98.8%

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

      10. associate-*r* 98.8%

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

      11. distribute-lft-out 98.8%

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

      12. *-lft-identity 98.8%

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

      13. distribute-rgt-in 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 72.6%

      \[\leadsto \frac{1}{\color{blue}{3 \cdot \frac{{x}^{2}}{s} + \left(4 \cdot s + 4 \cdot x\right)}} \]

   9. Taylor expanded in x around inf 72.6%

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

       1. associate-*r/ 72.6%

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

       2. associate-/l* 72.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{3}{\frac{s}{{x}^{2}}}}} \]

   11. Simplified 72.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{3}{\frac{s}{{x}^{2}}}}} \]

   12. Taylor expanded in s around 0 72.6%

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

       1. associate-*r/ 72.6%

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

       2. associate-*l/ 72.6%

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

       3. *-commutative 72.6%

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

   14. Simplified 72.6%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{2} \cdot \frac{3}{s}}\\ \end{array} \]

Alternative 8: 74.6% accurate, 5.6× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (* s (pow (+ (/ x s) 2.0) 2.0))))

C

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

Fortran

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(s * (Float32(Float32(x / s) + Float32(2.0)) ^ Float32(2.0))))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Applied egg-rr 59.2%

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

   1. associate-+r+ 59.2%

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

   2. *-lft-identity 59.2%

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

   3. distribute-rgt-in 59.2%

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

   4. +-commutative 59.2%

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

   5. *-commutative 59.2%

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

   6. fma-udef 59.2%

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

   7. distribute-rgt-in 59.2%

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

   8. *-rgt-identity 59.2%

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

   9. distribute-lft-in 59.2%

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

   10. associate-*r* 59.2%

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

   11. distribute-lft-out 59.3%

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

   12. *-lft-identity 59.3%

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

   13. distribute-rgt-in 59.3%

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

7. Simplified 59.3%

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

8. Taylor expanded in x around 0 70.0%

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

9. Final simplification 70.0%

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

Alternative 9: 45.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(s \cdot {x}^{-2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.999999987845058e-8)
   (/ 0.25 s)
   (* 0.3333333333333333 (* s (pow x -2.0)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.999999987845058e-8f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.3333333333333333f * (s * powf(x, -2.0f));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.999999987845058e-8) then
        tmp = 0.25e0 / s
    else
        tmp = 0.3333333333333333e0 * (s * (x ** (-2.0e0)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.999999987845058e-8))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.3333333333333333) * Float32(s * (x ^ Float32(-2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.999999987845058e-8))
		tmp = single(0.25) / s;
	else
		tmp = single(0.3333333333333333) * (s * (x ^ single(-2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999e-8

   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. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 34.5%

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

   if 1.99999999e-8 < 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%

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

   5. Applied egg-rr 98.8%

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

      1. associate-+r+ 98.8%

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

      2. *-lft-identity 98.8%

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

      3. distribute-rgt-in 98.8%

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

      4. +-commutative 98.8%

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

      5. *-commutative 98.8%

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

      6. fma-udef 98.8%

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

      7. distribute-rgt-in 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-in 98.8%

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

      10. associate-*r* 98.8%

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

      11. distribute-lft-out 98.8%

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

      12. *-lft-identity 98.8%

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

      13. distribute-rgt-in 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 72.6%

      \[\leadsto \frac{1}{\color{blue}{3 \cdot \frac{{x}^{2}}{s} + \left(4 \cdot s + 4 \cdot x\right)}} \]

   9. Taylor expanded in x around inf 72.6%

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

       1. associate-*r/ 72.6%

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

       2. associate-/l* 72.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{3}{\frac{s}{{x}^{2}}}}} \]

   11. Simplified 72.6%

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

       1. expm1-log1p-u 72.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{3}{\frac{s}{{x}^{2}}}}\right)\right)} \]

       2. expm1-udef 90.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{3}{\frac{s}{{x}^{2}}}}\right)} - 1} \]

       3. associate-/r/ 90.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{3} \cdot \frac{s}{{x}^{2}}}\right)} - 1 \]

       4. metadata-eval 90.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.3333333333333333} \cdot \frac{s}{{x}^{2}}\right)} - 1 \]

       5. div-inv 90.2%

          \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \color{blue}{\left(s \cdot \frac{1}{{x}^{2}}\right)}\right)} - 1 \]

       6. pow-flip 90.2%

          \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \left(s \cdot \color{blue}{{x}^{\left(-2\right)}}\right)\right)} - 1 \]

       7. metadata-eval 90.2%

          \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \left(s \cdot {x}^{\color{blue}{-2}}\right)\right)} - 1 \]

   13. Applied egg-rr 90.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \left(s \cdot {x}^{-2}\right)\right)} - 1} \]
   14. Step-by-step derivation

       1. expm1-def 69.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 \cdot \left(s \cdot {x}^{-2}\right)\right)\right)} \]

       2. expm1-log1p 69.8%

          \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(s \cdot {x}^{-2}\right)} \]

   15. Simplified 69.8%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(s \cdot {x}^{-2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(s \cdot {x}^{-2}\right)\\ \end{array} \]

Alternative 10: 45.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{s}{{x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.999999987845058e-8)
   (/ 0.25 s)
   (* 0.3333333333333333 (/ s (pow x 2.0)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.999999987845058e-8f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.3333333333333333f * (s / powf(x, 2.0f));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.999999987845058e-8) then
        tmp = 0.25e0 / s
    else
        tmp = 0.3333333333333333e0 * (s / (x ** 2.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.999999987845058e-8))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.3333333333333333) * Float32(s / (x ^ Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.999999987845058e-8))
		tmp = single(0.25) / s;
	else
		tmp = single(0.3333333333333333) * (s / (x ^ single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999e-8

   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. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 34.5%

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

   if 1.99999999e-8 < 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%

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

   5. Applied egg-rr 98.8%

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

      1. associate-+r+ 98.8%

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

      2. *-lft-identity 98.8%

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

      3. distribute-rgt-in 98.8%

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

      4. +-commutative 98.8%

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

      5. *-commutative 98.8%

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

      6. fma-udef 98.8%

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

      7. distribute-rgt-in 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-in 98.8%

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

      10. associate-*r* 98.8%

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

      11. distribute-lft-out 98.8%

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

      12. *-lft-identity 98.8%

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

      13. distribute-rgt-in 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 72.6%

      \[\leadsto \frac{1}{\color{blue}{3 \cdot \frac{{x}^{2}}{s} + \left(4 \cdot s + 4 \cdot x\right)}} \]

   9. Taylor expanded in x around inf 70.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{s}{{x}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{s}{{x}^{2}}\\ \end{array} \]

Alternative 11: 45.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{0.3333333333333333}{{x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.999999987845058e-8)
   (/ 0.25 s)
   (* s (/ 0.3333333333333333 (pow x 2.0)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.999999987845058e-8f) {
		tmp = 0.25f / s;
	} else {
		tmp = s * (0.3333333333333333f / powf(x, 2.0f));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.999999987845058e-8) then
        tmp = 0.25e0 / s
    else
        tmp = s * (0.3333333333333333e0 / (x ** 2.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.999999987845058e-8))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s * Float32(Float32(0.3333333333333333) / (x ^ Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.999999987845058e-8))
		tmp = single(0.25) / s;
	else
		tmp = s * (single(0.3333333333333333) / (x ^ single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999e-8

   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. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 34.5%

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

   if 1.99999999e-8 < 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%

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

   5. Applied egg-rr 98.8%

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

      1. associate-+r+ 98.8%

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

      2. *-lft-identity 98.8%

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

      3. distribute-rgt-in 98.8%

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

      4. +-commutative 98.8%

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

      5. *-commutative 98.8%

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

      6. fma-udef 98.8%

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

      7. distribute-rgt-in 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-in 98.8%

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

      10. associate-*r* 98.8%

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

      11. distribute-lft-out 98.8%

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

      12. *-lft-identity 98.8%

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

      13. distribute-rgt-in 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 72.6%

      \[\leadsto \frac{1}{\color{blue}{3 \cdot \frac{{x}^{2}}{s} + \left(4 \cdot s + 4 \cdot x\right)}} \]

   9. Taylor expanded in x around inf 72.6%

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

       1. associate-*r/ 72.6%

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

       2. associate-/l* 72.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{3}{\frac{s}{{x}^{2}}}}} \]

   11. Simplified 72.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{3}{\frac{s}{{x}^{2}}}}} \]

   12. Taylor expanded in s around 0 70.6%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{s}{{x}^{2}}} \]
   13. Step-by-step derivation

       1. associate-*r/ 70.6%

          \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot s}{{x}^{2}}} \]

       2. *-rgt-identity 70.6%

          \[\leadsto \frac{0.3333333333333333 \cdot s}{\color{blue}{{x}^{2} \cdot 1}} \]

       3. times-frac 70.6%

          \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{2}} \cdot \frac{s}{1}} \]

       4. /-rgt-identity 70.6%

          \[\leadsto \frac{0.3333333333333333}{{x}^{2}} \cdot \color{blue}{s} \]

   14. Simplified 70.6%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{2}} \cdot s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{0.3333333333333333}{{x}^{2}}\\ \end{array} \]

Alternative 12: 45.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot 0.3333333333333333}{{x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.999999987845058e-8)
   (/ 0.25 s)
   (/ (* s 0.3333333333333333) (pow x 2.0))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.999999987845058e-8f) {
		tmp = 0.25f / s;
	} else {
		tmp = (s * 0.3333333333333333f) / powf(x, 2.0f);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.999999987845058e-8) then
        tmp = 0.25e0 / s
    else
        tmp = (s * 0.3333333333333333e0) / (x ** 2.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.999999987845058e-8))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(s * Float32(0.3333333333333333)) / (x ^ Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.999999987845058e-8))
		tmp = single(0.25) / s;
	else
		tmp = (s * single(0.3333333333333333)) / (x ^ single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999e-8

   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. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 34.5%

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

   if 1.99999999e-8 < 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%

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

   5. Applied egg-rr 98.8%

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

      1. associate-+r+ 98.8%

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

      2. *-lft-identity 98.8%

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

      3. distribute-rgt-in 98.8%

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

      4. +-commutative 98.8%

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

      5. *-commutative 98.8%

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

      6. fma-udef 98.8%

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

      7. distribute-rgt-in 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-in 98.8%

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

      10. associate-*r* 98.8%

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

      11. distribute-lft-out 98.8%

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

      12. *-lft-identity 98.8%

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

      13. distribute-rgt-in 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 72.6%

      \[\leadsto \frac{1}{\color{blue}{3 \cdot \frac{{x}^{2}}{s} + \left(4 \cdot s + 4 \cdot x\right)}} \]

   9. Taylor expanded in x around inf 72.6%

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

       1. associate-*r/ 72.6%

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

       2. associate-/l* 72.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{3}{\frac{s}{{x}^{2}}}}} \]

   11. Simplified 72.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{3}{\frac{s}{{x}^{2}}}}} \]

   12. Taylor expanded in s around 0 70.6%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{s}{{x}^{2}}} \]
   13. Step-by-step derivation

       1. associate-*r/ 70.6%

          \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot s}{{x}^{2}}} \]

   14. Simplified 70.6%

       \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot s}{{x}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot 0.3333333333333333}{{x}^{2}}\\ \end{array} \]

Alternative 13: 51.0% accurate, 56.4× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (s * (4.0f + ((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 * (4.0e0 + ((x / s) * 4.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Applied egg-rr 59.2%

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

   1. associate-+r+ 59.2%

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

   2. *-lft-identity 59.2%

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

   3. distribute-rgt-in 59.2%

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

   4. +-commutative 59.2%

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

   5. *-commutative 59.2%

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

   6. fma-udef 59.2%

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

   7. distribute-rgt-in 59.2%

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

   8. *-rgt-identity 59.2%

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

   9. distribute-lft-in 59.2%

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

   10. associate-*r* 59.2%

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

   11. distribute-lft-out 59.3%

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

   12. *-lft-identity 59.3%

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

   13. distribute-rgt-in 59.3%

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

7. Simplified 59.3%

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

8. Taylor expanded in x around 0 49.7%

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

9. Final simplification 49.7%

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

Alternative 14: 29.8% accurate, 88.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Applied egg-rr 59.2%

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

   1. associate-+r+ 59.2%

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

   2. *-lft-identity 59.2%

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

   3. distribute-rgt-in 59.2%

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

   4. +-commutative 59.2%

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

   5. *-commutative 59.2%

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

   6. fma-udef 59.2%

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

   7. distribute-rgt-in 59.2%

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

   8. *-rgt-identity 59.2%

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

   9. distribute-lft-in 59.2%

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

   10. associate-*r* 59.2%

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

   11. distribute-lft-out 59.3%

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

   12. *-lft-identity 59.3%

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

   13. distribute-rgt-in 59.3%

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

7. Simplified 59.3%

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

8. Taylor expanded in s around inf 28.2%

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

   1. distribute-lft-out 28.2%

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

10. Simplified 28.2%

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

11. Final simplification 28.2%

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

Alternative 15: 29.1% accurate, 121.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.999999987845058e-8) (/ 0.25 s) (/ 0.25 x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.999999987845058e-8f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.25f / 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 <= 1.999999987845058e-8) then
        tmp = 0.25e0 / s
    else
        tmp = 0.25e0 / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.999999987845058e-8))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.25) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.999999987845058e-8))
		tmp = single(0.25) / s;
	else
		tmp = single(0.25) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999e-8

   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. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 34.5%

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

   if 1.99999999e-8 < 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%

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

   5. Applied egg-rr 98.8%

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

      1. associate-+r+ 98.8%

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

      2. *-lft-identity 98.8%

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

      3. distribute-rgt-in 98.8%

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

      4. +-commutative 98.8%

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

      5. *-commutative 98.8%

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

      6. fma-udef 98.8%

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

      7. distribute-rgt-in 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. distribute-lft-in 98.8%

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

      10. associate-*r* 98.8%

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

      11. distribute-lft-out 98.8%

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

      12. *-lft-identity 98.8%

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

      13. distribute-rgt-in 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in s around inf 11.5%

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

      1. distribute-lft-out 11.5%

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

   10. Simplified 11.5%

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

   11. Taylor expanded in s around 0 10.7%

       \[\leadsto \color{blue}{\frac{0.25}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{x}\\ \end{array} \]

Alternative 16: 29.4% accurate, 124.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Applied egg-rr 59.2%

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

   1. associate-+r+ 59.2%

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

   2. *-lft-identity 59.2%

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

   3. distribute-rgt-in 59.2%

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

   4. +-commutative 59.2%

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

   5. *-commutative 59.2%

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

   6. fma-udef 59.2%

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

   7. distribute-rgt-in 59.2%

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

   8. *-rgt-identity 59.2%

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

   9. distribute-lft-in 59.2%

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

   10. associate-*r* 59.2%

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

   11. distribute-lft-out 59.3%

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

   12. *-lft-identity 59.3%

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

   13. distribute-rgt-in 59.3%

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

7. Simplified 59.3%

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

8. Taylor expanded in s around inf 28.2%

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

   1. distribute-lft-out 28.2%

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

10. Simplified 28.2%

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

    1. expm1-log1p-u 26.4%

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

    2. expm1-udef 63.3%

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

    3. associate-/r* 63.3%

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

    4. metadata-eval 63.3%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.25}}{s + x}\right)} - 1 \]

    5. +-commutative 63.3%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{0.25}{\color{blue}{x + s}}\right)} - 1 \]

12. Applied egg-rr 63.3%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.25}{x + s}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 25.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.25}{x + s}\right)\right)} \]

    2. expm1-log1p 27.7%

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

    3. +-commutative 27.7%

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

14. Simplified 27.7%

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

15. Final simplification 27.7%

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

Alternative 17: 27.6% 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 99.8%

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

   1. *-commutative 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 25.7%

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

5. Final simplification 25.7%

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

Reproduce

herbie shell --seed 2023334 
(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))))))