Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 12.9s

Alternatives: 7

Speedup: N/A×

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: 99.0% 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: 99.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (* PI (/ 1.0 (- 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) * (1.0f / -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(pi) * Float32(Float32(1.0) / 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) * (single(1.0) / -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. Step-by-step derivation

   1. frac-2neg 98.9%

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

   2. div-inv 98.9%

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

   3. remove-double-neg 98.9%

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

5. Applied egg-rr 98.9%

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

6. Final simplification 98.9%

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

Alternative 2: 99.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

Alternative 3: 11.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \log \left(e^{\mathsf{fma}\left(u \cdot \pi, 0.5, \pi \cdot -0.25\right)}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* 4.0 (log (exp (fma (* u PI) 0.5 (* PI -0.25))))))

C

float code(float u, float s) {
	return 4.0f * logf(expf(fmaf((u * ((float) M_PI)), 0.5f, (((float) M_PI) * -0.25f))));
}

Julia

function code(u, s)
	return Float32(Float32(4.0) * log(exp(fma(Float32(u * Float32(pi)), Float32(0.5), Float32(Float32(pi) * Float32(-0.25))))))
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 10.8%

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

   1. associate--r+ 10.8%

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

   2. cancel-sign-sub-inv 10.8%

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

   3. distribute-rgt-out-- 10.8%

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

   4. *-commutative 10.8%

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

   5. metadata-eval 10.8%

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

   6. metadata-eval 10.8%

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

   7. *-commutative 10.8%

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

6. Simplified 10.8%

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

   1. add-log-exp 10.8%

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

   2. fma-def 10.8%

      \[\leadsto 4 \cdot \log \left(e^{\color{blue}{\mathsf{fma}\left(\pi \cdot u, 0.5, \pi \cdot -0.25\right)}}\right) \]

8. Applied egg-rr 10.8%

   \[\leadsto 4 \cdot \color{blue}{\log \left(e^{\mathsf{fma}\left(\pi \cdot u, 0.5, \pi \cdot -0.25\right)}\right)} \]

9. Final simplification 10.8%

   \[\leadsto 4 \cdot \log \left(e^{\mathsf{fma}\left(u \cdot \pi, 0.5, \pi \cdot -0.25\right)}\right) \]

Alternative 4: 11.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(\pi \cdot -0.25 + 0.5 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(u \cdot \pi\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* 4.0 (+ (* PI -0.25) (* 0.5 (expm1 (log1p (* u PI)))))))

C

float code(float u, float s) {
	return 4.0f * ((((float) M_PI) * -0.25f) + (0.5f * expm1f(log1pf((u * ((float) M_PI))))));
}

Julia

function code(u, s)
	return Float32(Float32(4.0) * Float32(Float32(Float32(pi) * Float32(-0.25)) + Float32(Float32(0.5) * expm1(log1p(Float32(u * 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 10.8%

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

   1. associate--r+ 10.8%

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

   2. cancel-sign-sub-inv 10.8%

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

   3. distribute-rgt-out-- 10.8%

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

   4. *-commutative 10.8%

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

   5. metadata-eval 10.8%

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

   6. metadata-eval 10.8%

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

   7. *-commutative 10.8%

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

6. Simplified 10.8%

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

   1. expm1-log1p-u 10.8%

      \[\leadsto 4 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot u\right)\right)} \cdot 0.5 + \pi \cdot -0.25\right) \]

8. Applied egg-rr 10.8%

   \[\leadsto 4 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot u\right)\right)} \cdot 0.5 + \pi \cdot -0.25\right) \]

9. Final simplification 10.8%

   \[\leadsto 4 \cdot \left(\pi \cdot -0.25 + 0.5 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(u \cdot \pi\right)\right)\right) \]

Alternative 5: 11.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = log(exp((single(pi) * (single(-1.0) + (u * single(2.0))))));
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. Step-by-step derivation

   1. frac-2neg 98.9%

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

   2. div-inv 98.9%

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

   3. remove-double-neg 98.9%

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

5. Applied egg-rr 98.9%

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

6. Taylor expanded in s around -inf 10.8%

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

   1. associate--r+ 10.8%

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

   2. cancel-sign-sub-inv 10.8%

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

   3. *-commutative 10.8%

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

   4. *-commutative 10.8%

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

   5. distribute-rgt-out-- 10.8%

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

   6. metadata-eval 10.8%

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

   7. metadata-eval 10.8%

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

   8. *-commutative 10.8%

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

   9. fma-udef 10.8%

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

   10. *-lft-identity 10.8%

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

   11. *-lft-identity 10.8%

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

   12. fma-udef 10.8%

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

   13. *-commutative 10.8%

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

   14. *-commutative 10.8%

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

   15. *-commutative 10.8%

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

8. Simplified 10.8%

   \[\leadsto \color{blue}{\pi \cdot \left(-1 + 2 \cdot u\right)} \]
9. Step-by-step derivation

   1. add-log-exp 10.8%

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

   2. *-commutative 10.8%

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

10. Applied egg-rr 10.8%

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

11. Final simplification 10.8%

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

Alternative 6: 11.5% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \left(-1 + u \cdot 2\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* PI (+ -1.0 (* u 2.0))))

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(pi) * (single(-1.0) + (u * single(2.0)));
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. Step-by-step derivation

   1. frac-2neg 98.9%

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

   2. div-inv 98.9%

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

   3. remove-double-neg 98.9%

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

5. Applied egg-rr 98.9%

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

6. Taylor expanded in s around -inf 10.8%

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

   1. associate--r+ 10.8%

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

   2. cancel-sign-sub-inv 10.8%

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

   3. *-commutative 10.8%

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

   4. *-commutative 10.8%

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

   5. distribute-rgt-out-- 10.8%

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

   6. metadata-eval 10.8%

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

   7. metadata-eval 10.8%

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

   8. *-commutative 10.8%

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

   9. fma-udef 10.8%

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

   10. *-lft-identity 10.8%

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

   11. *-lft-identity 10.8%

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

   12. fma-udef 10.8%

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

   13. *-commutative 10.8%

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

   14. *-commutative 10.8%

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

   15. *-commutative 10.8%

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

8. Simplified 10.8%

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

9. Final simplification 10.8%

   \[\leadsto \pi \cdot \left(-1 + u \cdot 2\right) \]

Alternative 7: 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 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 10.6%

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

   1. mul-1-neg 10.6%

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

6. Simplified 10.6%

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

7. Final simplification 10.6%

   \[\leadsto -\pi \]

Reproduce

herbie shell --seed 2023195 
(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))))