Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 15.5s

Alternatives: 6

Speedup: 1.3×

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, 1.3× speedup

Math

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

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(s * Float32(-log(Float32(Float32(Float32(1.0) / Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / 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 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 25.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 25.0%

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

   1. +-commutative 25.0%

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

   2. fma-define 25.0%

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

   3. associate--r+ 25.0%

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

   4. cancel-sign-sub-inv 25.0%

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

   5. distribute-rgt-out-- 25.0%

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

   6. *-commutative 25.0%

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

   7. metadata-eval 25.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot \color{blue}{0.5} + \left(-0.25\right) \cdot \pi}{s}, 1\right)\right) \]

   8. metadata-eval 25.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{-0.25} \cdot \pi}{s}, 1\right)\right) \]

   9. *-commutative 25.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{\pi \cdot -0.25}}{s}, 1\right)\right) \]

7. Simplified 25.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}, 1\right)\right)} \]

8. Taylor expanded in s around 0 25.2%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\log \left(-4 \cdot \left(-0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)\right)\right) + \left(-1 \cdot \log s + -0.25 \cdot \frac{s}{-0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)}\right)\right)} \]

9. Taylor expanded in u around 0 25.4%

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

10. Final simplification 25.4%

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

Alternative 3: 25.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 25.0%

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

   1. +-commutative 25.0%

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

   2. fma-define 25.0%

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

   3. associate--r+ 25.0%

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

   4. cancel-sign-sub-inv 25.0%

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

   5. distribute-rgt-out-- 25.0%

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

   6. *-commutative 25.0%

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

   7. metadata-eval 25.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot \color{blue}{0.5} + \left(-0.25\right) \cdot \pi}{s}, 1\right)\right) \]

   8. metadata-eval 25.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{-0.25} \cdot \pi}{s}, 1\right)\right) \]

   9. *-commutative 25.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{\pi \cdot -0.25}}{s}, 1\right)\right) \]

7. Simplified 25.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}, 1\right)\right)} \]

8. Taylor expanded in u around 0 25.1%

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

   1. clear-num 25.1%

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

   2. associate-/r/ 25.1%

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

10. Applied egg-rr 25.1%

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

11. Taylor expanded in s around 0 25.3%

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

    1. *-commutative 25.3%

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

    2. associate-*r* 25.3%

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

    3. mul-1-neg 25.3%

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

    4. log-rec 25.2%

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

    5. *-commutative 25.2%

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

    6. mul-1-neg 25.2%

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

    7. log-rec 25.3%

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

    8. mul-1-neg 25.3%

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

    9. +-commutative 25.3%

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

    10. distribute-neg-in 25.3%

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

    11. mul-1-neg 25.3%

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

    12. remove-double-neg 25.3%

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

    13. sub-neg 25.3%

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

    14. log-div 25.3%

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

13. Simplified 25.3%

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

14. Final simplification 25.3%

    \[\leadsto s \cdot \log \left(\frac{s}{\pi}\right) \]
15. Add Preprocessing

Alternative 4: 13.2% accurate, 72.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 25.0%

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

   1. +-commutative 25.0%

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

   2. fma-define 25.0%

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

   3. associate--r+ 25.0%

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

   4. cancel-sign-sub-inv 25.0%

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

   5. distribute-rgt-out-- 25.0%

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

   6. *-commutative 25.0%

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

   7. metadata-eval 25.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot \color{blue}{0.5} + \left(-0.25\right) \cdot \pi}{s}, 1\right)\right) \]

   8. metadata-eval 25.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{-0.25} \cdot \pi}{s}, 1\right)\right) \]

   9. *-commutative 25.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{\pi \cdot -0.25}}{s}, 1\right)\right) \]

7. Simplified 25.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}, 1\right)\right)} \]

8. Taylor expanded in u around 0 25.1%

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

9. Taylor expanded in s around inf 11.5%

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

    1. distribute-lft-neg-out 11.5%

       \[\leadsto \color{blue}{-s \cdot \frac{\pi}{s}} \]

    2. neg-sub0 11.5%

       \[\leadsto \color{blue}{0 - s \cdot \frac{\pi}{s}} \]

    3. div-inv 11.5%

       \[\leadsto 0 - s \cdot \color{blue}{\left(\pi \cdot \frac{1}{s}\right)} \]

    4. add-exp-log 11.5%

       \[\leadsto 0 - s \cdot \color{blue}{e^{\log \left(\pi \cdot \frac{1}{s}\right)}} \]

    5. log-prod 11.5%

       \[\leadsto 0 - s \cdot e^{\color{blue}{\log \pi + \log \left(\frac{1}{s}\right)}} \]

    6. neg-log 11.5%

       \[\leadsto 0 - s \cdot e^{\log \pi + \color{blue}{\left(-\log s\right)}} \]

    7. mul-1-neg 11.5%

       \[\leadsto 0 - s \cdot e^{\log \pi + \color{blue}{-1 \cdot \log s}} \]

    8. add-sqr-sqrt 11.5%

       \[\leadsto 0 - s \cdot e^{\log \pi + \color{blue}{\sqrt{-1 \cdot \log s} \cdot \sqrt{-1 \cdot \log s}}} \]

    9. sqrt-unprod 11.5%

       \[\leadsto 0 - s \cdot e^{\log \pi + \color{blue}{\sqrt{\left(-1 \cdot \log s\right) \cdot \left(-1 \cdot \log s\right)}}} \]

    10. mul-1-neg 11.5%

        \[\leadsto 0 - s \cdot e^{\log \pi + \sqrt{\color{blue}{\left(-\log s\right)} \cdot \left(-1 \cdot \log s\right)}} \]

    11. mul-1-neg 11.5%

        \[\leadsto 0 - s \cdot e^{\log \pi + \sqrt{\left(-\log s\right) \cdot \color{blue}{\left(-\log s\right)}}} \]

    12. sqr-neg 11.5%

        \[\leadsto 0 - s \cdot e^{\log \pi + \sqrt{\color{blue}{\log s \cdot \log s}}} \]

    13. sqrt-unprod -0.0%

        \[\leadsto 0 - s \cdot e^{\log \pi + \color{blue}{\sqrt{\log s} \cdot \sqrt{\log s}}} \]

    14. add-sqr-sqrt 13.4%

        \[\leadsto 0 - s \cdot e^{\log \pi + \color{blue}{\log s}} \]

    15. prod-exp 13.4%

        \[\leadsto 0 - s \cdot \color{blue}{\left(e^{\log \pi} \cdot e^{\log s}\right)} \]

    16. add-exp-log 13.4%

        \[\leadsto 0 - s \cdot \left(\color{blue}{\pi} \cdot e^{\log s}\right) \]

    17. add-exp-log 13.4%

        \[\leadsto 0 - s \cdot \left(\pi \cdot \color{blue}{s}\right) \]

11. Applied egg-rr 13.4%

    \[\leadsto \color{blue}{0 - s \cdot \left(\pi \cdot s\right)} \]
12. Step-by-step derivation

    1. neg-sub0 13.4%

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

    2. distribute-rgt-neg-in 13.4%

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

    3. distribute-lft-neg-in 13.4%

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

13. Simplified 13.4%

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

14. Final simplification 13.4%

    \[\leadsto s \cdot \left(s \cdot \left(-\pi\right)\right) \]
15. Add Preprocessing

Alternative 5: 11.4% accurate, 216.5× 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 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in u around 0 11.5%

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

   1. neg-mul-1 11.5%

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

7. Simplified 11.5%

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

8. Final simplification 11.5%

   \[\leadsto -\pi \]
9. Add Preprocessing

Alternative 6: 4.6% accurate, 433.0× 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(pi)
end

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 25.0%

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

   1. +-commutative 25.0%

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

   2. fma-define 25.0%

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

   3. associate--r+ 25.0%

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

   4. cancel-sign-sub-inv 25.0%

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

   5. distribute-rgt-out-- 25.0%

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

   6. *-commutative 25.0%

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

   7. metadata-eval 25.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot \color{blue}{0.5} + \left(-0.25\right) \cdot \pi}{s}, 1\right)\right) \]

   8. metadata-eval 25.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{-0.25} \cdot \pi}{s}, 1\right)\right) \]

   9. *-commutative 25.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{\pi \cdot -0.25}}{s}, 1\right)\right) \]

7. Simplified 25.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(-4, \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}, 1\right)\right)} \]

8. Taylor expanded in u around 0 25.1%

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

9. Taylor expanded in s around inf 11.5%

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

    1. clear-num 11.5%

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

    2. un-div-inv 11.5%

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

    3. add-sqr-sqrt -0.0%

       \[\leadsto \frac{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}{\frac{s}{\pi}} \]

    4. sqrt-unprod 7.9%

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

    5. sqr-neg 7.9%

       \[\leadsto \frac{\sqrt{\color{blue}{s \cdot s}}}{\frac{s}{\pi}} \]

    6. sqrt-unprod 4.6%

       \[\leadsto \frac{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}{\frac{s}{\pi}} \]

    7. add-sqr-sqrt 4.6%

       \[\leadsto \frac{\color{blue}{s}}{\frac{s}{\pi}} \]

11. Applied egg-rr 4.6%

    \[\leadsto \color{blue}{\frac{s}{\frac{s}{\pi}}} \]
12. Step-by-step derivation

    1. associate-/r/ 4.6%

       \[\leadsto \color{blue}{\frac{s}{s} \cdot \pi} \]

    2. *-inverses 4.6%

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

    3. *-lft-identity 4.6%

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

13. Simplified 4.6%

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

14. Final simplification 4.6%

    \[\leadsto \pi \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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))))