Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 11.9s

Alternatives: 4

Speedup: 1.0×

• unsoundness detected in the egraph

Specification

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
  (*
   (- s)
   (log
    (-
     (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
     1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)));
end

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
  (*
   (- s)
   (log
    (-
     (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
     1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)));
end

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-3.1415927410125732}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
  (*
   (- s)
   (log
    (-
     (/
      1.0
      (+
       (* u (- (/ 1.0 (+ 1.0 (exp (/ -3.1415927410125732 s)))) t_0))
       t_0))
     1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-3.1415927410125732f / s)))) - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-3.1415927410125732) / s)))) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((single(-3.1415927410125732) / s)))) - t_0)) + t_0)) - single(1.0)));
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

2. Evaluated real constant 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\color{blue}{-3.1415927410125732}}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
3. Add Preprocessing

Alternative 2: 37.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - t\_0\right) + t\_0} - 1\right) \end{array} \]

FPCore

(FPCore (u s)
  :precision binary32
  (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
  (* (- s) (log (- (/ 1.0 (+ (* u (- 0.5 t_0)) t_0)) 1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / ((u * (0.5f - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(0.5) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = -s * log(((single(1.0) / ((u * (single(0.5) - t_0)) + t_0)) - single(1.0)));
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

2. Taylor expanded in s around inf

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

4. Applied rewrites 37.7%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\color{blue}{0.5} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
5. Add Preprocessing

Alternative 3: 11.6% accurate, 10.0× speedup

Math

\[\mathsf{fma}\left(-1, \pi, 6.2831854820251465 \cdot u\right) \]

FPCore

(FPCore (u s)
  :precision binary32
  (fma -1.0 PI (* 6.2831854820251465 u)))

C

float code(float u, float s) {
	return fmaf(-1.0f, ((float) M_PI), (6.2831854820251465f * u));
}

Julia

function code(u, s)
	return fma(Float32(-1.0), Float32(pi), Float32(Float32(6.2831854820251465) * u))
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right) - \frac{1}{4} \cdot \pi\right)} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto 4 \cdot \color{blue}{\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

   2. lower--.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\frac{1}{4} \cdot \mathsf{PI}\left(\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \color{blue}{\frac{1}{4}} \cdot \mathsf{PI}\left(\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   6. lower-PI.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   7. lower-*.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   8. lower-PI.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   9. lower-*.f32 N/A

      \[\leadsto 4 \cdot \left(u \cdot \left(\frac{1}{4} \cdot \pi - \frac{-1}{4} \cdot \pi\right) - \frac{1}{4} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   10. lower-PI.f32 11.6%

       \[\leadsto 4 \cdot \left(u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) - 0.25 \cdot \pi\right) \]

4. Applied rewrites 11.6%

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) - 0.25 \cdot \pi\right)} \]

5. Evaluated real constant 11.6%

   \[\leadsto 4 \cdot \left(u \cdot 1.5707963705062866 - 0.25 \cdot \pi\right) \]

6. Taylor expanded in u around 0

   \[\leadsto -1 \cdot \pi + \color{blue}{\frac{13176795}{2097152} \cdot u} \]
7. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(-1, \mathsf{PI}\left(\right), \frac{13176795}{2097152} \cdot u\right) \]

   2. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(-1, \pi, \frac{13176795}{2097152} \cdot u\right) \]

   3. lower-*.f32 11.6%

      \[\leadsto \mathsf{fma}\left(-1, \pi, 6.2831854820251465 \cdot u\right) \]

8. Applied rewrites 11.6%

   \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\pi}, 6.2831854820251465 \cdot u\right) \]
9. Add Preprocessing

Alternative 4: 11.4% accurate, 90.2× speedup

Math

\[-3.1415927410125732 \]

FPCore

(FPCore (u s)
  :precision binary32
  -3.1415927410125732)

C

float code(float u, float s) {
	return -3.1415927410125732f;
}

Fortran

real(4) function code(u, s)
use fmin_fmax_functions
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = -3.1415927410125732e0
end function

Julia

function code(u, s)
	return Float32(-3.1415927410125732)
end

MATLAB

function tmp = code(u, s)
	tmp = single(-3.1415927410125732);
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

2. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \pi} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto -1 \cdot \color{blue}{\mathsf{PI}\left(\right)} \]

   2. lower-PI.f32 11.4%

      \[\leadsto -1 \cdot \pi \]

4. Applied rewrites 11.4%

   \[\leadsto \color{blue}{-1 \cdot \pi} \]

5. Evaluated real constant 11.4%

   \[\leadsto -3.1415927410125732 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025322 
(FPCore (u s)
  :name "Sample trimmed logistic on [-pi, pi]"
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0)) (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- s) (log (- (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) 1.0))))