Logistic distribution

Percentage Accurate: 99.6% → 99.2%

Time: 12.3s

Alternatives: 10

Speedup: 2.9×

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.2% accurate, 2.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ (/ 1.0 s) (* (+ 1.0 (exp (/ (- x) s))) (+ 1.0 (exp (/ (fabs x) s))))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / s) / ((1.0e0 + exp((-x / s))) * (1.0e0 + exp((abs(x) / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. neg-mul-1 99.5%

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

   3. times-frac 99.5%

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

   4. metadata-eval 99.5%

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

   5. metadata-eval 99.5%

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

   6. times-frac 99.5%

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

   7. neg-mul-1 99.5%

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

   8. *-un-lft-identity 99.5%

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

   9. distribute-frac-neg 99.5%

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

   10. rec-exp 99.4%

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

   11. add-sqr-sqrt 45.6%

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

   12. fabs-sqr 45.6%

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

   13. add-sqr-sqrt 96.1%

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

5. Applied egg-rr 96.1%

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

   1. rec-exp 96.1%

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

   2. distribute-neg-frac 96.1%

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

7. Simplified 96.1%

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

8. Taylor expanded in s around 0 96.1%

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

   1. associate-/r* 96.2%

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

   2. associate-*r/ 96.2%

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

   3. mul-1-neg 96.2%

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

10. Simplified 96.2%

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

11. Final simplification 96.2%

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

Alternative 2: 99.6% accurate, 2.9× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ 1.0 (* s (* (+ 1.0 (exp (/ (- x) s))) (+ 1.0 (exp (/ x s)))))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * ((1.0e0 + exp((-x / s))) * (1.0e0 + exp((x / s)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. neg-mul-1 99.5%

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

   3. times-frac 99.5%

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

   4. metadata-eval 99.5%

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

   5. metadata-eval 99.5%

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

   6. times-frac 99.5%

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

   7. neg-mul-1 99.5%

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

   8. *-un-lft-identity 99.5%

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

   9. distribute-frac-neg 99.5%

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

   10. rec-exp 99.4%

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

   11. add-sqr-sqrt 45.6%

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

   12. fabs-sqr 45.6%

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

   13. add-sqr-sqrt 96.1%

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

5. Applied egg-rr 96.1%

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

   1. rec-exp 96.1%

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

   2. distribute-neg-frac 96.1%

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

7. Simplified 96.1%

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

   1. add-sqr-sqrt 45.6%

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

   2. fabs-sqr 45.6%

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

   3. add-sqr-sqrt 99.5%

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

   4. expm1-log1p-u 99.4%

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

   5. expm1-udef 99.4%

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

9. Applied egg-rr 99.4%

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

    1. expm1-def 99.4%

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

    2. expm1-log1p 99.5%

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

11. Simplified 99.5%

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

12. Taylor expanded in s around 0 99.4%

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

    1. *-commutative 99.4%

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

    2. neg-mul-1 99.4%

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

14. Simplified 99.4%

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

15. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 2.9× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ 1.0 (* (+ 1.0 (exp (/ x s))) (+ s (* s (exp (/ (- x) s)))))))

C

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

Fortran

NOTE: x should be positive before calling this function
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

x = abs(x)
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

x = abs(x)
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 99.4%

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

3. Simplified 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. neg-mul-1 99.5%

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

   3. times-frac 99.5%

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

   4. metadata-eval 99.5%

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

   5. metadata-eval 99.5%

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

   6. times-frac 99.5%

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

   7. neg-mul-1 99.5%

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

   8. *-un-lft-identity 99.5%

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

   9. distribute-frac-neg 99.5%

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

   10. rec-exp 99.4%

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

   11. add-sqr-sqrt 45.6%

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

   12. fabs-sqr 45.6%

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

   13. add-sqr-sqrt 96.1%

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

5. Applied egg-rr 96.1%

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

   1. rec-exp 96.1%

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

   2. distribute-neg-frac 96.1%

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

7. Simplified 96.1%

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

   1. add-sqr-sqrt 45.6%

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

   2. fabs-sqr 45.6%

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

   3. add-sqr-sqrt 99.5%

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

   4. expm1-log1p-u 99.4%

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

   5. expm1-udef 99.4%

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

9. Applied egg-rr 99.4%

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

    1. expm1-def 99.4%

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

    2. expm1-log1p 99.5%

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

11. Simplified 99.5%

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

12. Taylor expanded in x around inf 99.4%

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

    1. mul-1-neg 99.4%

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

    2. exp-neg 99.4%

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

14. Applied egg-rr 99.4%

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

    1. rec-exp 99.4%

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

    2. distribute-neg-frac 99.4%

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

16. Simplified 99.4%

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

17. Final simplification 99.4%

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

Alternative 4: 95.1% accurate, 5.5× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ (+ s s) (* (exp (/ x s)) (+ s s)))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((s + s) + (exp((x / s)) * (s + s)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around inf 94.8%

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

   1. distribute-lft-in 94.8%

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

   2. *-rgt-identity 94.8%

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

   3. fma-udef 94.8%

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

   4. *-rgt-identity 94.8%

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

   5. fma-udef 94.8%

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

   6. *-rgt-identity 94.8%

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

   7. add-sqr-sqrt 43.9%

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

   8. fabs-sqr 43.9%

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

   9. add-sqr-sqrt 60.5%

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

6. Applied egg-rr 60.5%

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

7. Final simplification 60.5%

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

Alternative 5: 95.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{0.5}{s \cdot \left(1 + e^{\frac{x}{s}}\right)} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 0.5 (* s (+ 1.0 (exp (/ x s))))))

C

x = abs(x);
float code(float x, float s) {
	return 0.5f / (s * (1.0f + expf((x / s))));
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / (s * (1.0e0 + exp((x / s))))
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(0.5) / Float32(s * Float32(Float32(1.0) + exp(Float32(x / s)))))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = single(0.5) / (s * (single(1.0) + exp((x / s))));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around inf 94.8%

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

   1. add-log-exp 78.1%

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

   2. associate-/r* 78.1%

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

   3. fma-udef 78.1%

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

   4. *-rgt-identity 78.1%

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

   5. add-sqr-sqrt 37.7%

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

   6. fabs-sqr 37.7%

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

   7. add-sqr-sqrt 43.6%

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

6. Applied egg-rr 43.6%

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

7. Taylor expanded in s around 0 60.5%

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

8. Final simplification 60.5%

   \[\leadsto \frac{0.5}{s \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

Alternative 6: 65.1% accurate, 56.4× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 1.0 (+ (* s 4.0) (/ (* x x) s))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((s * 4.0e0) + ((x * x) / s))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around -inf 42.6%

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

   1. +-commutative 42.6%

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

   2. mul-1-neg 42.6%

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

   3. distribute-lft1-in 64.5%

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

   4. metadata-eval 64.5%

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

   5. associate-*r/ 64.5%

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

   6. mul-1-neg 64.5%

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

   7. remove-double-neg 64.5%

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

   8. associate-+r+ 64.5%

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

6. Simplified 64.9%

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

7. Final simplification 64.9%

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

Alternative 7: 30.0% accurate, 68.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{1}{s \cdot 4 + x \cdot 2} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 1.0 (+ (* s 4.0) (* x 2.0))))

C

x = abs(x);
float code(float x, float s) {
	return 1.0f / ((s * 4.0f) + (x * 2.0f));
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((s * 4.0e0) + (x * 2.0e0))
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(s * Float32(4.0)) + Float32(x * Float32(2.0))))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = single(1.0) / ((s * single(4.0)) + (x * single(2.0)));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around inf 94.8%

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

   1. distribute-lft-in 94.8%

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

   2. *-rgt-identity 94.8%

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

   3. fma-udef 94.8%

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

   4. *-rgt-identity 94.8%

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

   5. fma-udef 94.8%

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

   6. *-rgt-identity 94.8%

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

   7. add-sqr-sqrt 43.9%

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

   8. fabs-sqr 43.9%

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

   9. add-sqr-sqrt 60.5%

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

6. Applied egg-rr 60.5%

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

7. Taylor expanded in s around inf 29.3%

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

8. Final simplification 29.3%

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

Alternative 8: 50.7% accurate, 68.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\frac{0.5}{s}}{\frac{x}{s} + 2} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ (/ 0.5 s) (+ (/ x s) 2.0)))

C

x = abs(x);
float code(float x, float s) {
	return (0.5f / s) / ((x / s) + 2.0f);
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (0.5e0 / s) / ((x / s) + 2.0e0)
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(Float32(0.5) / s) / Float32(Float32(x / s) + Float32(2.0)))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = (single(0.5) / s) / ((x / s) + single(2.0));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around inf 94.8%

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

   1. add-log-exp 78.1%

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

   2. associate-/r* 78.1%

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

   3. fma-udef 78.1%

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

   4. *-rgt-identity 78.1%

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

   5. add-sqr-sqrt 37.7%

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

   6. fabs-sqr 37.7%

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

   7. add-sqr-sqrt 43.6%

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

6. Applied egg-rr 43.6%

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

7. Taylor expanded in s around 0 60.5%

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

   1. associate-/r* 60.5%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{s}}{1 + e^{\frac{x}{s}}}} \]

9. Simplified 60.5%

   \[\leadsto \color{blue}{\frac{\frac{0.5}{s}}{1 + e^{\frac{x}{s}}}} \]

10. Taylor expanded in x around 0 50.5%

    \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{2 + \frac{x}{s}}} \]

11. Final simplification 50.5%

    \[\leadsto \frac{\frac{0.5}{s}}{\frac{x}{s} + 2} \]

Alternative 9: 85.4% accurate, 121.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 2.0000000390829628e-24) (/ 0.25 s) 0.0))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 2.0000000390829628e-24f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.0f;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 2.0000000390829628e-24) then
        tmp = 0.25e0 / s
    else
        tmp = 0.0e0
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.0000000390829628e-24))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(0.0);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.0000000390829628e-24))
		tmp = single(0.25) / s;
	else
		tmp = single(0.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.00000004e-24

   1. Initial program 99.1%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in s around inf 37.9%

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

   if 2.00000004e-24 < 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%

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

   4. Taylor expanded in s around inf 96.5%

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

      1. add-log-exp 92.3%

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

      2. associate-/r* 92.3%

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

      3. fma-udef 92.3%

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

      4. *-rgt-identity 92.3%

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

      5. add-sqr-sqrt 92.3%

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

      6. fabs-sqr 92.3%

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

      7. add-sqr-sqrt 92.3%

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

   6. Applied egg-rr 92.3%

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

   7. Taylor expanded in s around inf 90.7%

      \[\leadsto \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 27.6% accurate, 206.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 0.25 s))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / s
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around inf 27.3%

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

5. Final simplification 27.3%

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

Reproduce

herbie shell --seed 2023308 
(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))))))