Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 9.6s

Alternatives: 7

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, 1.4× 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 98.9%

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

   1. distribute-lft-neg-out 98.9%

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

   2. distribute-rgt-neg-in 98.9%

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

   3. sub-neg 98.9%

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

3. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 2: 25.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{s \cdot \left(-s\right)}{\pi} - s \cdot \mathsf{fma}\left(-1, \log s, \log \pi\right) \end{array} \]

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(Float32(s * Float32(-s)) / Float32(pi)) - Float32(s * fma(Float32(-1.0), log(s), log(Float32(pi)))))
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. Step-by-step derivation

   1. distribute-lft-neg-out 98.9%

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

   2. distribute-rgt-neg-in 98.9%

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

   3. sub-neg 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in s around -inf 24.7%

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

   1. +-commutative 24.7%

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

   2. fma-def 24.7%

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

   3. associate--r+ 24.7%

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

   4. cancel-sign-sub-inv 24.7%

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

   5. distribute-rgt-out-- 24.7%

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

   6. *-commutative 24.7%

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

   7. metadata-eval 24.7%

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

   8. metadata-eval 24.7%

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

   9. *-commutative 24.7%

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

6. Simplified 24.7%

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

7. Taylor expanded in u around 0 24.9%

   \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(4, \frac{\color{blue}{0.25 \cdot \pi}}{s}, 1\right)\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 24.9%

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

9. Simplified 24.9%

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

10. Taylor expanded in s around 0 25.0%

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

    1. distribute-lft-out 25.0%

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

    2. unpow2 25.0%

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

    3. fma-def 25.0%

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

12. Simplified 25.0%

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

13. Final simplification 25.0%

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

Alternative 3: 25.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * (log(s) - (log(single(pi)) + (s / single(pi))));
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. Step-by-step derivation

   1. distribute-lft-neg-out 98.9%

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

   2. distribute-rgt-neg-in 98.9%

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

   3. sub-neg 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in s around -inf 24.7%

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

   1. +-commutative 24.7%

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

   2. fma-def 24.7%

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

   3. associate--r+ 24.7%

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

   4. cancel-sign-sub-inv 24.7%

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

   5. distribute-rgt-out-- 24.7%

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

   6. *-commutative 24.7%

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

   7. metadata-eval 24.7%

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

   8. metadata-eval 24.7%

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

   9. *-commutative 24.7%

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

6. Simplified 24.7%

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

7. Taylor expanded in u around 0 24.9%

   \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(4, \frac{\color{blue}{0.25 \cdot \pi}}{s}, 1\right)\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 24.9%

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

9. Simplified 24.9%

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

10. Taylor expanded in s around 0 25.0%

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

11. Final simplification 25.0%

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

Alternative 4: 25.2% accurate, 3.6× 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 98.9%

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

   1. distribute-lft-neg-out 98.9%

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

   2. distribute-rgt-neg-in 98.9%

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

   3. sub-neg 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in s around -inf 24.7%

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

   1. +-commutative 24.7%

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

   2. fma-def 24.7%

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

   3. associate--r+ 24.7%

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

   4. cancel-sign-sub-inv 24.7%

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

   5. distribute-rgt-out-- 24.7%

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

   6. *-commutative 24.7%

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

   7. metadata-eval 24.7%

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

   8. metadata-eval 24.7%

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

   9. *-commutative 24.7%

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

6. Simplified 24.7%

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

7. Taylor expanded in u around 0 24.9%

   \[\leadsto s \cdot \left(-\log \left(\mathsf{fma}\left(4, \frac{\color{blue}{0.25 \cdot \pi}}{s}, 1\right)\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 24.9%

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

9. Simplified 24.9%

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

10. Taylor expanded in s around 0 25.0%

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

11. Taylor expanded in s around 0 25.0%

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

    1. mul-1-neg 25.0%

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

    2. neg-mul-1 25.0%

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

    3. +-commutative 25.0%

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

    4. sub-neg 25.0%

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

    5. log-div 24.9%

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

    6. distribute-rgt-neg-in 24.9%

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

    7. log-div 25.0%

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

    8. sub-neg 25.0%

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

    9. +-commutative 25.0%

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

    10. distribute-neg-in 25.0%

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

    11. remove-double-neg 25.0%

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

    12. sub-neg 25.0%

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

    13. log-div 25.0%

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

13. Simplified 25.0%

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

14. Final simplification 25.0%

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

Alternative 5: 14.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;s \cdot 0\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\pi \cdot \left(u \cdot 0.5 + -0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 9.999999682655225e-21)
   (* s 0.0)
   (* 4.0 (* PI (+ (* u 0.5) -0.25)))))

C

float code(float u, float s) {
	float tmp;
	if (s <= 9.999999682655225e-21f) {
		tmp = s * 0.0f;
	} else {
		tmp = 4.0f * (((float) M_PI) * ((u * 0.5f) + -0.25f));
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(9.999999682655225e-21))
		tmp = Float32(s * Float32(0.0));
	else
		tmp = Float32(Float32(4.0) * Float32(Float32(pi) * Float32(Float32(u * Float32(0.5)) + Float32(-0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	tmp = single(0.0);
	if (s <= single(9.999999682655225e-21))
		tmp = s * single(0.0);
	else
		tmp = single(4.0) * (single(pi) * ((u * single(0.5)) + single(-0.25)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 9.99999968e-21

   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. Step-by-step derivation

      1. distribute-lft-neg-out 99.0%

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

      2. distribute-rgt-neg-in 99.0%

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

      3. sub-neg 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in s around inf 14.0%

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

   if 9.99999968e-21 < s

   1. Initial program 98.8%

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

      1. distribute-lft-neg-out 98.8%

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

      2. distribute-rgt-neg-in 98.8%

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

      3. sub-neg 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in s around inf 14.3%

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

      1. associate--r+ 14.3%

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

      2. cancel-sign-sub-inv 14.3%

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

      3. distribute-rgt-out-- 14.3%

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

      4. *-commutative 14.3%

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

      5. metadata-eval 14.3%

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

      6. metadata-eval 14.3%

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

      7. *-commutative 14.3%

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

   6. Simplified 14.3%

      \[\leadsto \color{blue}{4 \cdot \left(\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25\right)} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 14.3%

         \[\leadsto 4 \cdot \left(\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot -0.25\right) \]

      2. pow3 14.3%

         \[\leadsto 4 \cdot \left(\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot -0.25\right) \]

   8. Applied egg-rr 14.3%

      \[\leadsto 4 \cdot \left(\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot -0.25\right) \]

   9. Taylor expanded in u around 0 14.3%

      \[\leadsto 4 \cdot \color{blue}{\left(-0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative 14.3%

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

       2. *-commutative 14.3%

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

       3. *-commutative 14.3%

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

       4. *-commutative 14.3%

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

       5. associate-*r* 14.3%

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

       6. distribute-lft-out 14.3%

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

       7. *-commutative 14.3%

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

   11. Simplified 14.3%

       \[\leadsto 4 \cdot \color{blue}{\left(\pi \cdot \left(0.5 \cdot u + -0.25\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;s \cdot 0\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\pi \cdot \left(u \cdot 0.5 + -0.25\right)\right)\\ \end{array} \]

Alternative 6: 11.5% 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 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. Step-by-step derivation

   1. distribute-lft-neg-out 98.9%

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

   2. distribute-rgt-neg-in 98.9%

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

   3. sub-neg 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in u around 0 10.5%

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

   1. mul-1-neg 10.5%

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

6. Simplified 10.5%

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

7. Final simplification 10.5%

   \[\leadsto -\pi \]

Alternative 7: 10.3% accurate, 243.3× speedup

Math

\[\begin{array}{l} \\ s \cdot 0 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* s 0.0))

C

float code(float u, float s) {
	return s * 0.0f;
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * 0.0e0
end function

Julia

function code(u, s)
	return Float32(s * Float32(0.0))
end

MATLAB

function tmp = code(u, s)
	tmp = s * single(0.0);
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. Step-by-step derivation

   1. distribute-lft-neg-out 98.9%

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

   2. distribute-rgt-neg-in 98.9%

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

   3. sub-neg 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in s around inf 11.0%

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

5. Final simplification 11.0%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023229 
(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))))