Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 4.1s

Alternatives: 11

Speedup: 1.1×

Specification

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

Math

\[\begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (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)
use fmin_fmax_functions
    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

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

Math

\[\begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (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)
use fmin_fmax_functions
    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, 1.1× speedup

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

Math

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

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (let* ((t_0 (exp (/ (- (fabs x)) s))))
  (/ (* (pow (- -1.0 t_0) -2.0) t_0) s)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

Alternative 2: 95.0% accurate, 1.4× speedup

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

Math

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

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/
 -1.0
 (*
  (* (- -1.0 (exp (/ (- (fabs x)) s))) (+ s s))
  (exp (/ (fabs x) s)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

4. Applied rewrites 95.0%

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

6. Applied rewrites 95.0%

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

Alternative 3: 95.0% accurate, 1.4× speedup

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

Math

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

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (let* ((t_0 (exp (/ (- (fabs x)) s))))
  (/ (/ t_0 (+ 1.0 t_0)) (+ s s))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

4. Applied rewrites 95.0%

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

6. Applied rewrites 95.0%

   \[\leadsto \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}{s + s} \]
7. Add Preprocessing

Alternative 4: 95.0% accurate, 1.4× speedup

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

Math

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

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (let* ((t_0 (exp (/ (- (fabs x)) s))))
  (/ t_0 (* (+ s s) (+ 1.0 t_0)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

4. Applied rewrites 95.0%

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

6. Applied rewrites 95.0%

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

Alternative 5: 94.6% accurate, 2.5× speedup

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

Math

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

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ -1.0 (* (* -4.0 s) (exp (/ (fabs x) s)))))

C

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

Fortran

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (-1.0e0) / (((-4.0e0) * s) * exp((abs(x) / s)))
end function

Julia

function code(x, s)
	return Float32(Float32(-1.0) / Float32(Float32(Float32(-4.0) * s) * exp(Float32(abs(x) / s))))
end

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

4. Applied rewrites 95.0%

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

6. Applied rewrites 95.0%

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

7. Taylor expanded in s around inf

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

9. Applied rewrites 94.6%

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

Alternative 6: 94.6% accurate, 2.7× speedup

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

Math

\[\frac{e^{\frac{-\left|x\right|}{s}}}{4 \cdot s} \]

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ (exp (/ (- (fabs x)) s)) (* 4.0 s)))

C

float code(float x, float s) {
	return expf((-fabsf(x) / s)) / (4.0f * s);
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{4 \cdot s} \]
3. Step-by-step derivation

4. Applied rewrites 94.6%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{4 \cdot s} \]
5. Add Preprocessing

Alternative 7: 56.1% accurate, 2.9× speedup

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

Math

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

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ -1.0 (* (- (* 4.0 s)) (+ 1.0 (/ (sqrt (* x x)) s)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{4 \cdot s} \]
3. Step-by-step derivation

4. Applied rewrites 94.6%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{4 \cdot s} \]

6. Applied rewrites 94.6%

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

7. Taylor expanded in s around inf

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

9. Applied rewrites 49.7%

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

11. Applied rewrites 56.1%

    \[\leadsto \frac{-1}{\left(-4 \cdot s\right) \cdot \left(1 + \frac{\sqrt{x \cdot x}}{s}\right)} \]
12. Add Preprocessing

Alternative 8: 49.8% accurate, 3.0× speedup

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

Math

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

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ -1.0 (* (- (* 4.0 s)) (* (+ s (fabs x)) (/ 1.0 s)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{4 \cdot s} \]
3. Step-by-step derivation

4. Applied rewrites 94.6%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{4 \cdot s} \]

6. Applied rewrites 94.6%

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

7. Taylor expanded in s around inf

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

9. Applied rewrites 49.7%

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

11. Applied rewrites 49.8%

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

Alternative 9: 49.7% accurate, 3.5× speedup

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

Math

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

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ -1.0 (* (- (* 4.0 s)) (+ 1.0 (/ (fabs x) s)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{4 \cdot s} \]
3. Step-by-step derivation

4. Applied rewrites 94.6%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{4 \cdot s} \]

6. Applied rewrites 94.6%

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

7. Taylor expanded in s around inf

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

9. Applied rewrites 49.7%

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

Alternative 10: 27.3% accurate, 8.5× speedup

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

Math

\[\frac{-0.125}{-0.5 \cdot s} \]

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ -0.125 (* -0.5 s)))

C

float code(float x, float s) {
	return -0.125f / (-0.5f * s);
}

Fortran

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (-0.125e0) / ((-0.5e0) * s)
end function

Julia

function code(x, s)
	return Float32(Float32(-0.125) / Float32(Float32(-0.5) * s))
end

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{4}}{s} \]
3. Step-by-step derivation

4. Applied rewrites 27.3%

   \[\leadsto \frac{0.25}{s} \]
5. Step-by-step derivation

6. Applied rewrites 27.3%

   \[\leadsto 0.125 \cdot \frac{1}{0.5 \cdot s} \]
7. Step-by-step derivation

8. Applied rewrites 27.3%

   \[\leadsto \frac{-0.125}{-0.5 \cdot s} \]
9. Add Preprocessing

Alternative 11: 27.3% accurate, 13.8× speedup

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

Math

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

FPCore

(FPCore (x s)
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 0.25 s))

C

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

Fortran

real(4) function code(x, s)
use fmin_fmax_functions
    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 99.5%

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

2. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{4}}{s} \]
3. Step-by-step derivation

4. Applied rewrites 27.3%

   \[\leadsto \frac{0.25}{s} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2026070 
(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))))))