Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 18.4s

Alternatives: 13

Speedup: 1.0×

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

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

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: 98.9% accurate, 1.0× speedup

Math

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

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: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ t_1 := 1 + t\_0\\ t_2 := -1 - t\_0\\ t_3 := e^{\frac{\pi}{-s}}\\ t_4 := \frac{u}{1 + t\_3} + \left(\frac{u}{t\_2} + \frac{-1}{t\_2}\right)\\ \left(-s\right) \cdot \log \left(\frac{-1 + {t\_4}^{-3}}{1 + \frac{1 + \frac{-1}{\frac{u}{-1 - t\_3} + \left(\frac{u}{t\_1} + \frac{-1}{t\_1}\right)}}{t\_4}}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s)))
        (t_1 (+ 1.0 t_0))
        (t_2 (- -1.0 t_0))
        (t_3 (exp (/ PI (- s))))
        (t_4 (+ (/ u (+ 1.0 t_3)) (+ (/ u t_2) (/ -1.0 t_2)))))
   (*
    (- s)
    (log
     (/
      (+ -1.0 (pow t_4 -3.0))
      (+
       1.0
       (/
        (+ 1.0 (/ -1.0 (+ (/ u (- -1.0 t_3)) (+ (/ u t_1) (/ -1.0 t_1)))))
        t_4)))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	float t_1 = 1.0f + t_0;
	float t_2 = -1.0f - t_0;
	float t_3 = expf((((float) M_PI) / -s));
	float t_4 = (u / (1.0f + t_3)) + ((u / t_2) + (-1.0f / t_2));
	return -s * logf(((-1.0f + powf(t_4, -3.0f)) / (1.0f + ((1.0f + (-1.0f / ((u / (-1.0f - t_3)) + ((u / t_1) + (-1.0f / t_1))))) / t_4))));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	t_2 = Float32(Float32(-1.0) - t_0)
	t_3 = exp(Float32(Float32(pi) / Float32(-s)))
	t_4 = Float32(Float32(u / Float32(Float32(1.0) + t_3)) + Float32(Float32(u / t_2) + Float32(Float32(-1.0) / t_2)))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(-1.0) + (t_4 ^ Float32(-3.0))) / Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(u / Float32(Float32(-1.0) - t_3)) + Float32(Float32(u / t_1) + Float32(Float32(-1.0) / t_1))))) / t_4)))))
end

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s));
	t_1 = single(1.0) + t_0;
	t_2 = single(-1.0) - t_0;
	t_3 = exp((single(pi) / -s));
	t_4 = (u / (single(1.0) + t_3)) + ((u / t_2) + (single(-1.0) / t_2));
	tmp = -s * log(((single(-1.0) + (t_4 ^ single(-3.0))) / (single(1.0) + ((single(1.0) + (single(-1.0) / ((u / (single(-1.0) - t_3)) + ((u / t_1) + (single(-1.0) / t_1))))) / t_4))));
end

Derivation

1. Initial program 98.9%

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

   1. lift--.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \color{blue}{\left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right)} \]

   2. sub-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \color{blue}{\left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   3. lift-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\color{blue}{\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   4. inv-pow N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\color{blue}{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{-1}} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   5. sqr-pow N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\color{blue}{{\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{\left(\frac{-1}{2}\right)}} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left({\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)}^{\left(\frac{-1}{2}\right)} + \color{blue}{-1}\right) \]

4. Applied rewrites 98.7%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left({\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-0.5}, {\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-0.5}, -1\right)\right)} \]

6. Applied rewrites 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ t_1 := -1 - t\_0\\ t_2 := \frac{u}{1 + e^{\frac{\pi}{-s}}}\\ \left(-s\right) \cdot \log \left(-1 + \frac{-1}{\left({\left(t\_2 + \frac{u}{t\_1}\right)}^{2} - {\left(1 + t\_0\right)}^{-2}\right) \cdot \frac{-1}{t\_2 + \frac{u - -1}{t\_1}}}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s)))
        (t_1 (- -1.0 t_0))
        (t_2 (/ u (+ 1.0 (exp (/ PI (- s)))))))
   (*
    (- s)
    (log
     (+
      -1.0
      (/
       -1.0
       (*
        (- (pow (+ t_2 (/ u t_1)) 2.0) (pow (+ 1.0 t_0) -2.0))
        (/ -1.0 (+ t_2 (/ (- u -1.0) t_1))))))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	float t_1 = -1.0f - t_0;
	float t_2 = u / (1.0f + expf((((float) M_PI) / -s)));
	return -s * logf((-1.0f + (-1.0f / ((powf((t_2 + (u / t_1)), 2.0f) - powf((1.0f + t_0), -2.0f)) * (-1.0f / (t_2 + ((u - -1.0f) / t_1)))))));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	t_1 = Float32(Float32(-1.0) - t_0)
	t_2 = Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s)))))
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(-1.0) / Float32(Float32((Float32(t_2 + Float32(u / t_1)) ^ Float32(2.0)) - (Float32(Float32(1.0) + t_0) ^ Float32(-2.0))) * Float32(Float32(-1.0) / Float32(t_2 + Float32(Float32(u - Float32(-1.0)) / t_1))))))))
end

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s));
	t_1 = single(-1.0) - t_0;
	t_2 = u / (single(1.0) + exp((single(pi) / -s)));
	tmp = -s * log((single(-1.0) + (single(-1.0) / ((((t_2 + (u / t_1)) ^ single(2.0)) - ((single(1.0) + t_0) ^ single(-2.0))) * (single(-1.0) / (t_2 + ((u - single(-1.0)) / t_1)))))));
end

Derivation

1. Initial program 98.9%

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

   1. lift-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}}} - 1\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right)} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   3. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}{s}}} - \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}\right) \cdot u} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   4. lower-fma.f32 98.9

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 98.8%

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

7. Final simplification 98.8%

   \[\leadsto \left(-s\right) \cdot \log \left(-1 + \frac{-1}{\left({\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{u}{-1 - e^{\frac{\pi}{s}}}\right)}^{2} - {\left(1 + e^{\frac{\pi}{s}}\right)}^{-2}\right) \cdot \frac{-1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{u - -1}{-1 - e^{\frac{\pi}{s}}}}}\right) \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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))))