Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 14.7s

Alternatives: 8

Speedup: 2.0×

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: 99.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{x\_m}{-s}}\\ \frac{\frac{t\_0}{s}}{{\left(t\_0 + 1\right)}^{2}} \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ x_m (- s))))) (/ (/ t_0 s) (pow (+ t_0 1.0) 2.0))))

C

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

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((x_m / -s))
    code = (t_0 / s) / ((t_0 + 1.0e0) ** 2.0e0)
end function

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	t_0 = exp((x_m / -s));
	tmp = (t_0 / s) / ((t_0 + single(1.0)) ^ single(2.0));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 99.6%

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

7. Simplified 62.9%

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

Alternative 2: 96.1% accurate, 5.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left(1 - \frac{x\_m}{s}\right)\\ \frac{\frac{e^{\frac{x\_m}{-s}}}{s}}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (- 1.0 (/ x_m s)))))
   (/ (/ (exp (/ x_m (- s))) s) (* t_0 t_0))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 99.6%

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

7. Simplified 62.9%

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

8. Taylor expanded in x around 0 60.8%

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

   1. neg-mul-1 60.8%

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

   2. unsub-neg 60.8%

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

10. Simplified 60.8%

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

    1. unpow2 60.8%

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

    2. +-commutative 60.8%

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

    3. +-commutative 60.8%

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

12. Applied egg-rr 60.8%

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

Alternative 3: 96.1% accurate, 5.2× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (/ (exp (/ x_m (- s))) s) (* (+ 1.0 (- 1.0 (/ x_m s))) (- 2.0 (/ x_m s)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 99.6%

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

7. Simplified 62.9%

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

8. Taylor expanded in x around 0 60.8%

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

   1. neg-mul-1 60.8%

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

   2. unsub-neg 60.8%

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

10. Simplified 60.8%

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

    1. unpow2 60.8%

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

    2. +-commutative 60.8%

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

    3. +-commutative 60.8%

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

12. Applied egg-rr 60.8%

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

13. Taylor expanded in x around 0 60.8%

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

    1. mul-1-neg 60.8%

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

    2. sub-neg 60.8%

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

15. Simplified 60.8%

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

16. Final simplification 60.8%

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

Alternative 4: 94.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{e^{\frac{x\_m}{-s}}}{s}}{4} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ (/ (exp (/ x_m (- s))) s) 4.0))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (expf((x_m / -s)) / s) / 4.0f;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = (exp((x_m / -s)) / s) / 4.0e0
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(exp(Float32(x_m / Float32(-s))) / s) / Float32(4.0))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (exp((x_m / -s)) / s) / single(4.0);
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 99.6%

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

7. Simplified 62.9%

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

8. Taylor expanded in x around 0 60.2%

   \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{s}}{\color{blue}{4}} \]
9. Add Preprocessing

Alternative 5: 94.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{0.25}{s}}{e^{\frac{x\_m}{s}}} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ (/ 0.25 s) (exp (/ x_m s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (0.25f / s) / expf((x_m / s));
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 99.6%

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

7. Simplified 62.9%

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

8. Taylor expanded in x around 0 60.2%

   \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{s}}{\color{blue}{4}} \]
9. Step-by-step derivation

   1. *-un-lft-identity 60.2%

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

   2. div-inv 60.2%

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

   3. clear-num 60.2%

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

   4. metadata-eval 60.2%

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

   5. associate-*l/ 60.2%

      \[\leadsto 1 \cdot \color{blue}{\frac{1 \cdot 0.25}{\frac{s}{e^{\frac{x}{-s}}}}} \]

   6. metadata-eval 60.2%

      \[\leadsto 1 \cdot \frac{\color{blue}{0.25}}{\frac{s}{e^{\frac{x}{-s}}}} \]

   7. div-inv 60.2%

      \[\leadsto 1 \cdot \frac{0.25}{\color{blue}{s \cdot \frac{1}{e^{\frac{x}{-s}}}}} \]

   8. frac-2neg 60.2%

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

   9. frac-2neg 60.2%

      \[\leadsto 1 \cdot \frac{0.25}{s \cdot \frac{1}{e^{\color{blue}{\frac{x}{-s}}}}} \]

   10. add-sqr-sqrt -0.0%

       \[\leadsto 1 \cdot \frac{0.25}{s \cdot \frac{1}{e^{\frac{x}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}}} \]

   11. sqrt-unprod 62.3%

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

   12. sqr-neg 62.3%

       \[\leadsto 1 \cdot \frac{0.25}{s \cdot \frac{1}{e^{\frac{x}{\sqrt{\color{blue}{s \cdot s}}}}}} \]

   13. sqrt-unprod 64.2%

       \[\leadsto 1 \cdot \frac{0.25}{s \cdot \frac{1}{e^{\frac{x}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}}} \]

   14. add-sqr-sqrt 64.2%

       \[\leadsto 1 \cdot \frac{0.25}{s \cdot \frac{1}{e^{\frac{x}{\color{blue}{s}}}}} \]

   15. exp-neg 64.2%

       \[\leadsto 1 \cdot \frac{0.25}{s \cdot \color{blue}{e^{-\frac{x}{s}}}} \]

   16. distribute-frac-neg2 64.2%

       \[\leadsto 1 \cdot \frac{0.25}{s \cdot e^{\color{blue}{\frac{x}{-s}}}} \]

   17. frac-2neg 64.2%

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

   18. frac-2neg 64.2%

       \[\leadsto 1 \cdot \frac{0.25}{s \cdot e^{\color{blue}{\frac{x}{-s}}}} \]

   19. add-sqr-sqrt -0.0%

       \[\leadsto 1 \cdot \frac{0.25}{s \cdot e^{\frac{x}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}} \]

   20. sqrt-unprod 58.4%

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

   21. sqr-neg 58.4%

       \[\leadsto 1 \cdot \frac{0.25}{s \cdot e^{\frac{x}{\sqrt{\color{blue}{s \cdot s}}}}} \]

   22. sqrt-unprod 60.2%

       \[\leadsto 1 \cdot \frac{0.25}{s \cdot e^{\frac{x}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}} \]

10. Applied egg-rr 60.2%

    \[\leadsto \color{blue}{1 \cdot \frac{0.25}{s \cdot e^{\frac{x}{s}}}} \]
11. Step-by-step derivation

    1. *-lft-identity 60.2%

       \[\leadsto \color{blue}{\frac{0.25}{s \cdot e^{\frac{x}{s}}}} \]

    2. associate-/r* 60.2%

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

12. Simplified 60.2%

    \[\leadsto \color{blue}{\frac{\frac{0.25}{s}}{e^{\frac{x}{s}}}} \]
13. Add Preprocessing

Alternative 6: 88.9% accurate, 41.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{0.25 \cdot \left(s + x\_m \cdot 0.5\right) + x\_m \cdot -0.125}{s}}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (/ (+ (* 0.25 (+ s (* x_m 0.5))) (* x_m -0.125)) s) s))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (((0.25f * (s + (x_m * 0.5f))) + (x_m * -0.125f)) / s) / s;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = (((0.25e0 * (s + (x_m * 0.5e0))) + (x_m * (-0.125e0))) / s) / s
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(Float32(Float32(0.25) * Float32(s + Float32(x_m * Float32(0.5)))) + Float32(x_m * Float32(-0.125))) / s) / s)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (((single(0.25) * (s + (x_m * single(0.5)))) + (x_m * single(-0.125))) / s) / s;
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

   1. associate-/r* 99.6%

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

   2. +-commutative 99.6%

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

   3. div-inv 99.6%

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

   4. fma-define 99.6%

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

   5. rec-exp 99.6%

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

   6. distribute-frac-neg 99.6%

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

   7. associate-/r* 99.6%

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

   8. div-inv 98.7%

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

6. Applied egg-rr 67.0%

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

   1. associate-*r/ 67.4%

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

   2. times-frac 67.0%

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

   3. rem-exp-log 67.0%

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

   4. +-commutative 67.0%

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

   5. log1p-undefine 66.9%

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

   6. exp-diff 87.3%

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

   7. associate-*r/ 87.7%

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

   8. *-rgt-identity 87.7%

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

8. Simplified 87.7%

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

9. Taylor expanded in s around inf 65.8%

   \[\leadsto \color{blue}{\frac{\left(0.5 \cdot e^{-\log 2} + 0.5 \cdot \frac{e^{-\log 2} \cdot \left(x - 0.5 \cdot x\right)}{s}\right) - 0.25 \cdot \frac{x \cdot e^{-\log 2}}{s}}{s}} \]
10. Step-by-step derivation

11. Simplified 30.3%

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

12. Taylor expanded in s around 0 90.8%

    \[\leadsto \frac{\color{blue}{\frac{\left(0.25 \cdot s + 0.25 \cdot \left(x + -0.5 \cdot x\right)\right) - 0.125 \cdot x}{s}}}{s} \]
13. Step-by-step derivation

    1. cancel-sign-sub-inv 90.8%

       \[\leadsto \frac{\frac{\color{blue}{\left(0.25 \cdot s + 0.25 \cdot \left(x + -0.5 \cdot x\right)\right) + \left(-0.125\right) \cdot x}}{s}}{s} \]

    2. distribute-lft-out 90.8%

       \[\leadsto \frac{\frac{\color{blue}{0.25 \cdot \left(s + \left(x + -0.5 \cdot x\right)\right)} + \left(-0.125\right) \cdot x}{s}}{s} \]

    3. distribute-rgt1-in 90.8%

       \[\leadsto \frac{\frac{0.25 \cdot \left(s + \color{blue}{\left(-0.5 + 1\right) \cdot x}\right) + \left(-0.125\right) \cdot x}{s}}{s} \]

    4. metadata-eval 90.8%

       \[\leadsto \frac{\frac{0.25 \cdot \left(s + \color{blue}{0.5} \cdot x\right) + \left(-0.125\right) \cdot x}{s}}{s} \]

    5. *-commutative 90.8%

       \[\leadsto \frac{\frac{0.25 \cdot \left(s + \color{blue}{x \cdot 0.5}\right) + \left(-0.125\right) \cdot x}{s}}{s} \]

    6. metadata-eval 90.8%

       \[\leadsto \frac{\frac{0.25 \cdot \left(s + x \cdot 0.5\right) + \color{blue}{-0.125} \cdot x}{s}}{s} \]

    7. *-commutative 90.8%

       \[\leadsto \frac{\frac{0.25 \cdot \left(s + x \cdot 0.5\right) + \color{blue}{x \cdot -0.125}}{s}}{s} \]

14. Simplified 90.8%

    \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot \left(s + x \cdot 0.5\right) + x \cdot -0.125}{s}}}{s} \]
15. Add Preprocessing

Alternative 7: 84.8% accurate, 77.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 2.0000000072549875e-15) (/ 0.25 s) 0.0))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 2.0000000072549875e-15f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.0f;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 2.0000000072549875e-15) then
        tmp = 0.25e0 / s
    else
        tmp = 0.0e0
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(2.0000000072549875e-15))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(0.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(2.0000000072549875e-15))
		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 < 2.00000001e-15

   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 99.9%

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

   5. Taylor expanded in s around inf 42.0%

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

   if 2.00000001e-15 < x

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

      1. associate-/r* 99.0%

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

      2. +-commutative 99.0%

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

      3. div-inv 99.0%

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

      4. fma-define 99.0%

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

      5. rec-exp 99.0%

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

      6. distribute-frac-neg 99.0%

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

      7. associate-/r* 99.0%

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

      8. div-inv 97.7%

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

   6. Applied egg-rr 8.3%

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

      1. associate-*r/ 8.3%

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

      2. times-frac 8.4%

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

      3. rem-exp-log 8.3%

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

      4. +-commutative 8.3%

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

      5. log1p-undefine 8.3%

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

      6. exp-diff 62.1%

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

      7. associate-*r/ 62.1%

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

      8. *-rgt-identity 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in s around inf 56.2%

      \[\leadsto \color{blue}{\frac{\left(0.5 \cdot e^{-\log 2} + 0.5 \cdot \frac{e^{-\log 2} \cdot \left(x - 0.5 \cdot x\right)}{s}\right) - 0.25 \cdot \frac{x \cdot e^{-\log 2}}{s}}{s}} \]
   10. Step-by-step derivation

   11. Simplified 7.3%

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

   12. Taylor expanded in s around 0 91.0%

       \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot \left(x + -0.5 \cdot x\right) - 0.125 \cdot x}{s}}}{s} \]
   13. Step-by-step derivation

       1. +-commutative 91.0%

          \[\leadsto \frac{\frac{0.25 \cdot \color{blue}{\left(-0.5 \cdot x + x\right)} - 0.125 \cdot x}{s}}{s} \]

       2. *-commutative 91.0%

          \[\leadsto \frac{\frac{0.25 \cdot \left(\color{blue}{x \cdot -0.5} + x\right) - 0.125 \cdot x}{s}}{s} \]

       3. fma-undefine 91.0%

          \[\leadsto \frac{\frac{0.25 \cdot \color{blue}{\mathsf{fma}\left(x, -0.5, x\right)} - 0.125 \cdot x}{s}}{s} \]

       4. div-sub 55.6%

          \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot \mathsf{fma}\left(x, -0.5, x\right)}{s} - \frac{0.125 \cdot x}{s}}}{s} \]

       5. *-commutative 55.6%

          \[\leadsto \frac{\frac{0.25 \cdot \mathsf{fma}\left(x, -0.5, x\right)}{s} - \frac{\color{blue}{x \cdot 0.125}}{s}}{s} \]

       6. metadata-eval 55.6%

          \[\leadsto \frac{\frac{0.25 \cdot \mathsf{fma}\left(x, -0.5, x\right)}{s} - \frac{x \cdot \color{blue}{\left(-0.125 + 0.25\right)}}{s}}{s} \]

       7. distribute-lft-out 55.6%

          \[\leadsto \frac{\frac{0.25 \cdot \mathsf{fma}\left(x, -0.5, x\right)}{s} - \frac{\color{blue}{x \cdot -0.125 + x \cdot 0.25}}{s}}{s} \]

       8. metadata-eval 55.6%

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

       9. associate-*l* 55.6%

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

       10. distribute-rgt-in 55.6%

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

       11. fma-undefine 55.6%

           \[\leadsto \frac{\frac{0.25 \cdot \mathsf{fma}\left(x, -0.5, x\right)}{s} - \frac{0.25 \cdot \color{blue}{\mathsf{fma}\left(x, -0.5, x\right)}}{s}}{s} \]

       12. +-inverses 91.0%

           \[\leadsto \frac{\color{blue}{0}}{s} \]

   14. Simplified 91.0%

       \[\leadsto \frac{\color{blue}{0}}{s} \]

   15. Taylor expanded in s around 0 91.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 73.2% accurate, 620.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 0.0)

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 0.0f;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(0.0)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

   1. associate-/r* 99.6%

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

   2. +-commutative 99.6%

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

   3. div-inv 99.6%

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

   4. fma-define 99.6%

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

   5. rec-exp 99.6%

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

   6. distribute-frac-neg 99.6%

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

   7. associate-/r* 99.6%

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

   8. div-inv 98.7%

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

6. Applied egg-rr 67.0%

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

   1. associate-*r/ 67.4%

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

   2. times-frac 67.0%

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

   3. rem-exp-log 67.0%

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

   4. +-commutative 67.0%

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

   5. log1p-undefine 66.9%

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

   6. exp-diff 87.3%

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

   7. associate-*r/ 87.7%

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

   8. *-rgt-identity 87.7%

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

8. Simplified 87.7%

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

9. Taylor expanded in s around inf 65.8%

   \[\leadsto \color{blue}{\frac{\left(0.5 \cdot e^{-\log 2} + 0.5 \cdot \frac{e^{-\log 2} \cdot \left(x - 0.5 \cdot x\right)}{s}\right) - 0.25 \cdot \frac{x \cdot e^{-\log 2}}{s}}{s}} \]
10. Step-by-step derivation

11. Simplified 30.3%

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

12. Taylor expanded in s around 0 70.1%

    \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot \left(x + -0.5 \cdot x\right) - 0.125 \cdot x}{s}}}{s} \]
13. Step-by-step derivation

    1. +-commutative 70.1%

       \[\leadsto \frac{\frac{0.25 \cdot \color{blue}{\left(-0.5 \cdot x + x\right)} - 0.125 \cdot x}{s}}{s} \]

    2. *-commutative 70.1%

       \[\leadsto \frac{\frac{0.25 \cdot \left(\color{blue}{x \cdot -0.5} + x\right) - 0.125 \cdot x}{s}}{s} \]

    3. fma-undefine 70.1%

       \[\leadsto \frac{\frac{0.25 \cdot \color{blue}{\mathsf{fma}\left(x, -0.5, x\right)} - 0.125 \cdot x}{s}}{s} \]

    4. div-sub 46.6%

       \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot \mathsf{fma}\left(x, -0.5, x\right)}{s} - \frac{0.125 \cdot x}{s}}}{s} \]

    5. *-commutative 46.6%

       \[\leadsto \frac{\frac{0.25 \cdot \mathsf{fma}\left(x, -0.5, x\right)}{s} - \frac{\color{blue}{x \cdot 0.125}}{s}}{s} \]

    6. metadata-eval 46.6%

       \[\leadsto \frac{\frac{0.25 \cdot \mathsf{fma}\left(x, -0.5, x\right)}{s} - \frac{x \cdot \color{blue}{\left(-0.125 + 0.25\right)}}{s}}{s} \]

    7. distribute-lft-out 46.6%

       \[\leadsto \frac{\frac{0.25 \cdot \mathsf{fma}\left(x, -0.5, x\right)}{s} - \frac{\color{blue}{x \cdot -0.125 + x \cdot 0.25}}{s}}{s} \]

    8. metadata-eval 46.6%

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

    9. associate-*l* 46.6%

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

    10. distribute-rgt-in 46.6%

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

    11. fma-undefine 46.6%

        \[\leadsto \frac{\frac{0.25 \cdot \mathsf{fma}\left(x, -0.5, x\right)}{s} - \frac{0.25 \cdot \color{blue}{\mathsf{fma}\left(x, -0.5, x\right)}}{s}}{s} \]

    12. +-inverses 70.1%

        \[\leadsto \frac{\color{blue}{0}}{s} \]

14. Simplified 70.1%

    \[\leadsto \frac{\color{blue}{0}}{s} \]

15. Taylor expanded in s around 0 70.1%

    \[\leadsto \color{blue}{0} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(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))))))