Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 32.8s

Alternatives: 8

Speedup: 1.4×

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

   1. un-div-inv 99.1%

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

   2. +-commutative 99.1%

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

   3. sub-neg 99.1%

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

   4. flip-+ 99.1%

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

5. Applied egg-rr 99.1%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

7. Applied egg-rr 99.1%

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

   1. unpow-1 99.1%

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

   2. +-commutative 99.1%

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

   3. +-commutative 99.1%

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

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

Alternative 2: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \left(\log \left(\sqrt{\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1}\right) \cdot 2\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (*
   (log
    (sqrt
     (+
      (/
       1.0
       (+
        (/ u (+ 1.0 (exp (/ PI (- s)))))
        (/ (- 1.0 u) (+ 1.0 (exp (/ PI s))))))
      -1.0)))
   2.0)))

C

float code(float u, float s) {
	return -s * (logf(sqrtf(((1.0f / ((u / (1.0f + expf((((float) M_PI) / -s)))) + ((1.0f - u) / (1.0f + expf((((float) M_PI) / s)))))) + -1.0f))) * 2.0f);
}

Julia

function code(u, s)
	return Float32(Float32(-s) * Float32(log(sqrt(Float32(Float32(Float32(1.0) / Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))) + Float32(-1.0)))) * Float32(2.0)))
end

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

   1. +-commutative 99.1%

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

   2. un-div-inv 99.1%

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

   3. +-commutative 99.1%

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

   4. sub-neg 99.1%

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

   5. expm1-log1p-u 98.8%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right)\right)}\right) \]

5. Applied egg-rr 98.8%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. expm1-log1p-u 99.1%

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

   2. add-log-exp 21.7%

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

   3. +-commutative 21.7%

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

7. Applied egg-rr 21.7%

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

   1. rem-log-exp 99.1%

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

   2. add-cube-cbrt 98.8%

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

   3. fma-def 98.8%

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

9. Applied egg-rr 99.1%

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

    1. +-commutative 99.1%

       \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(\sqrt[3]{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{\color{blue}{e^{\frac{\pi}{s}} + 1}}\right)}^{-2}}, \frac{1}{\sqrt[3]{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}, -1\right)\right)\right) \]

    2. +-commutative 99.1%

       \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(\sqrt[3]{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{e^{\frac{\pi}{s}} + 1}\right)}^{-2}}, \frac{1}{\sqrt[3]{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{\color{blue}{e^{\frac{\pi}{s}} + 1}}}}, -1\right)\right)\right) \]

11. Simplified 99.1%

    \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{e^{\frac{\pi}{s}} + 1}\right)}^{-2}}, \frac{1}{\sqrt[3]{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{e^{\frac{\pi}{s}} + 1}}}, -1\right)\right)}\right) \]

13. Applied egg-rr 99.1%

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

14. Final simplification 99.1%

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

Alternative 3: 98.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \mathsf{log1p}\left(-1 + \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log1p
   (+
    -1.0
    (+
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (- 1.0 u) (+ 1.0 (exp (/ PI s))))))
     -1.0)))))

C

float code(float u, float s) {
	return -s * log1pf((-1.0f + ((1.0f / ((u / (1.0f + expf((((float) M_PI) / -s)))) + ((1.0f - u) / (1.0f + expf((((float) M_PI) / s)))))) + -1.0f)));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log1p(Float32(Float32(-1.0) + Float32(Float32(Float32(1.0) / Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))) + Float32(-1.0)))))
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

   1. +-commutative 99.1%

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

   2. un-div-inv 99.1%

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

   3. +-commutative 99.1%

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

   4. sub-neg 99.1%

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

   5. expm1-log1p-u 98.8%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right)\right)}\right) \]

5. Applied egg-rr 98.8%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. expm1-log1p-u 99.1%

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

   2. add-log-exp 21.7%

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

   3. +-commutative 21.7%

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

7. Applied egg-rr 21.7%

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

   1. rem-log-exp 99.1%

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

   2. add-cube-cbrt 98.8%

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

   3. fma-def 98.8%

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

9. Applied egg-rr 99.1%

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

    1. +-commutative 99.1%

       \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(\sqrt[3]{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{\color{blue}{e^{\frac{\pi}{s}} + 1}}\right)}^{-2}}, \frac{1}{\sqrt[3]{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}, -1\right)\right)\right) \]

    2. +-commutative 99.1%

       \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(\sqrt[3]{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{e^{\frac{\pi}{s}} + 1}\right)}^{-2}}, \frac{1}{\sqrt[3]{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{\color{blue}{e^{\frac{\pi}{s}} + 1}}}}, -1\right)\right)\right) \]

11. Simplified 99.1%

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

13. Applied egg-rr 99.1%

    \[\leadsto s \cdot \left(-\color{blue}{\mathsf{log1p}\left(\left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right) - 1\right)}\right) \]

14. Final simplification 99.1%

    \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(-1 + \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right) \]

Alternative 4: 98.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 5: 98.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 6: 25.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\log s - \log \pi\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* s (- (log s) (log PI))))

C

float code(float u, float s) {
	return s * (logf(s) - logf(((float) M_PI)));
}

Julia

function code(u, s)
	return Float32(s * Float32(log(s) - log(Float32(pi))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * (log(s) - log(single(pi)));
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around -inf 25.1%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)}\right) \]
5. Step-by-step derivation

   1. +-commutative 25.1%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s} + 1\right)}\right) \]

   2. fma-def 25.1%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}, 1\right)\right)}\right) \]

6. Simplified 25.1%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(-0.25 \cdot u + 0.25\right) + \left(\pi \cdot u\right) \cdot -0.25}{s}, 1\right)\right)}\right) \]

7. Taylor expanded in s around 0 25.1%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \left(\log \left(4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) + \pi \cdot \left(0.25 + -0.25 \cdot u\right)\right)\right) + -1 \cdot \log s\right)\right)} \]

8. Taylor expanded in u around 0 25.3%

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

   1. neg-mul-1 25.3%

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

   2. unsub-neg 25.3%

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

10. Simplified 25.3%

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

11. Final simplification 25.3%

    \[\leadsto s \cdot \left(\log s - \log \pi\right) \]

Alternative 7: 25.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (- s) (log1p (/ PI s))))

C

float code(float u, float s) {
	return -s * log1pf((((float) M_PI) / s));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log1p(Float32(Float32(pi) / s)))
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around -inf 25.1%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)}\right) \]
5. Step-by-step derivation

   1. +-commutative 25.1%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s} + 1\right)}\right) \]

   2. fma-def 25.1%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}, 1\right)\right)}\right) \]

6. Simplified 25.1%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(-0.25 \cdot u + 0.25\right) + \left(\pi \cdot u\right) \cdot -0.25}{s}, 1\right)\right)}\right) \]

7. Taylor expanded in u around 0 25.2%

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

   1. associate-*r* 25.2%

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

   2. neg-mul-1 25.2%

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

   3. log1p-def 25.2%

      \[\leadsto \left(-s\right) \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

9. Simplified 25.2%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

10. Final simplification 25.2%

    \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

Alternative 8: 11.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ -\pi \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u around 0 11.2%

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

   1. neg-mul-1 11.2%

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

6. Simplified 11.2%

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

7. Final simplification 11.2%

   \[\leadsto -\pi \]

Reproduce

herbie shell --seed 2023336 
(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))))