Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 14.1s

Alternatives: 7

Speedup: 1.1×

• unsoundness detected in the egraph

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
  (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

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
  (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: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{-\left|x\right|}{s}\\ t_1 := \mathsf{expm1}\left(t\_0\right)\\ \frac{e^{t\_0}}{\mathsf{fma}\left(t\_1 - -1, \left(3 + t\_1\right) \cdot s, s\right)} \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  (let* ((t_0 (/ (- (fabs x)) s)) (t_1 (expm1 t_0)))
  (/ (exp t_0) (fma (- t_1 -1.0) (* (+ 3.0 t_1) s) s))))

C

float code(float x, float s) {
	float t_0 = -fabsf(x) / s;
	float t_1 = expm1f(t_0);
	return expf(t_0) / fmaf((t_1 - -1.0f), ((3.0f + t_1) * s), s);
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-abs(x)) / s)
	t_1 = expm1(t_0)
	return Float32(exp(t_0) / fma(Float32(t_1 - Float32(-1.0)), Float32(Float32(Float32(3.0) + t_1) * 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. Applied rewrites 99.5%

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

5. Applied rewrites 99.6%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\mathsf{fma}\left(\mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right) - -1, \left(3 + \mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right)\right) \cdot s, s\right)}} \]
6. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{-\left|x\right|}{s}\\ t_1 := \mathsf{expm1}\left(t\_0\right)\\ \frac{e^{t\_0}}{\mathsf{fma}\left(3 + t\_1, \mathsf{fma}\left(t\_1, s, s\right), s\right)} \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  (let* ((t_0 (/ (- (fabs x)) s)) (t_1 (expm1 t_0)))
  (/ (exp t_0) (fma (+ 3.0 t_1) (fma t_1 s s) s))))

C

float code(float x, float s) {
	float t_0 = -fabsf(x) / s;
	float t_1 = expm1f(t_0);
	return expf(t_0) / fmaf((3.0f + t_1), fmaf(t_1, s, s), s);
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-abs(x)) / s)
	t_1 = expm1(t_0)
	return Float32(exp(t_0) / fma(Float32(Float32(3.0) + t_1), fma(t_1, s, 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. Applied rewrites 99.5%

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

5. Applied rewrites 99.6%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\mathsf{fma}\left(3 + \mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right), \mathsf{fma}\left(\mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right), s, s\right), s\right)}} \]
6. Add Preprocessing

Alternative 3: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{-\left|x\right|}{s}\\ t_1 := -2 - \mathsf{expm1}\left(t\_0\right)\\ \frac{e^{t\_0}}{\left(t\_1 \cdot t\_1\right) \cdot s} \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  (let* ((t_0 (/ (- (fabs x)) s)) (t_1 (- -2.0 (expm1 t_0))))
  (/ (exp t_0) (* (* t_1 t_1) s))))

C

float code(float x, float s) {
	float t_0 = -fabsf(x) / s;
	float t_1 = -2.0f - expm1f(t_0);
	return expf(t_0) / ((t_1 * t_1) * s);
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-abs(x)) / s)
	t_1 = Float32(Float32(-2.0) - expm1(t_0))
	return Float32(exp(t_0) / Float32(Float32(t_1 * t_1) * 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. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(\left(-2 - \mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right)\right) \cdot \left(-2 - \mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right)\right)\right) \cdot s}} \]
6. Add Preprocessing

Alternative 4: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right)\\ t_1 := t\_0 - -2\\ \frac{t\_0 - -1}{\left(t\_1 \cdot s\right) \cdot t\_1} \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  (let* ((t_0 (expm1 (/ (- (fabs x)) s))) (t_1 (- t_0 -2.0)))
  (/ (- t_0 -1.0) (* (* t_1 s) t_1))))

C

float code(float x, float s) {
	float t_0 = expm1f((-fabsf(x) / s));
	float t_1 = t_0 - -2.0f;
	return (t_0 - -1.0f) / ((t_1 * s) * t_1);
}

Julia

function code(x, s)
	t_0 = expm1(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 - Float32(-2.0))
	return Float32(Float32(t_0 - Float32(-1.0)) / Float32(Float32(t_1 * s) * t_1))
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. Applied rewrites 98.5%

   \[\leadsto \frac{\color{blue}{3 - \left(3 - \left(1 - \left(-\mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right)\right)\right)\right)}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

5. Applied rewrites 98.6%

   \[\leadsto \frac{3 - \left(3 - \left(1 - \left(-\mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right)\right)\right)\right)}{\left(s \cdot \left(1 + \color{blue}{\left(3 - \left(3 - \left(1 - \left(-\mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right)\right)\right)\right)\right)}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

7. Applied rewrites 98.6%

   \[\leadsto \frac{3 - \left(3 - \left(1 - \left(-\mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right)\right)\right)\right)}{\left(s \cdot \left(1 + \left(3 - \left(3 - \left(1 - \left(-\mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right)\right)\right)\right)\right)\right)\right) \cdot \left(1 + \color{blue}{\left(3 - \left(3 - \left(1 - \left(-\mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right)\right)\right)\right)\right)}\right)} \]

9. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right) - -1}{\left(\left(\mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right) - -2\right) \cdot s\right) \cdot \left(\mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right) - -2\right)}} \]
10. Add Preprocessing

Alternative 5: 96.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{-\left|x\right|}{s}\\ \frac{e^{t\_0}}{\mathsf{fma}\left(\mathsf{expm1}\left(t\_0\right) - -1, 3 \cdot s, s\right)} \end{array} \]

FPCore

(FPCore (x s)
  :precision binary32
  (let* ((t_0 (/ (- (fabs x)) s)))
  (/ (exp t_0) (fma (- (expm1 t_0) -1.0) (* 3.0 s) s))))

C

float code(float x, float s) {
	float t_0 = -fabsf(x) / s;
	return expf(t_0) / fmaf((expm1f(t_0) - -1.0f), (3.0f * s), s);
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-abs(x)) / s)
	return Float32(exp(t_0) / fma(Float32(expm1(t_0) - Float32(-1.0)), Float32(Float32(3.0) * s), s))
end

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.6%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\mathsf{fma}\left(\mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right) - -1, \left(3 + \mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right)\right) \cdot s, s\right)}} \]

6. Taylor expanded in s around inf

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\mathsf{fma}\left(\mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right) - -1, \color{blue}{3} \cdot s, s\right)} \]
7. Step-by-step derivation

8. Applied rewrites 96.5%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\mathsf{fma}\left(\mathsf{expm1}\left(\frac{-\left|x\right|}{s}\right) - -1, \color{blue}{3} \cdot s, s\right)} \]
9. Add Preprocessing

Alternative 6: 95.1% accurate, 2.7× speedup

Math

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

FPCore

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

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

   1. lower-*.f32 95.1%

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

4. Applied rewrites 95.1%

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

Alternative 7: 26.7% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((single(1.0) - single(-1.0)) * (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

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

   1. lower-*.f32 95.4%

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

4. Applied rewrites 95.4%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 25.2%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 26.7%

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

    1. lift-*.f32 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f32 26.7%

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

    4. lift-+.f32 N/A

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

    5. +-commutative N/A

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

    6. add-flip N/A

       \[\leadsto \frac{1}{\color{blue}{\left(1 - \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(2 \cdot s\right)} \]

    7. metadata-eval N/A

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

    8. lower--.f32 26.7%

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

    9. lift-*.f32 N/A

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

    10. count-2-rev N/A

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

    11. lower-+.f32 26.7%

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

12. Applied rewrites 26.7%

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

Reproduce

herbie shell --seed 2025322 
(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))))))