Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 12.4s

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

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     -1.0
     (/
      1.0
      (-
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (+ 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)))) - ((u + -1.0f) / (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(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) - Float32(Float32(u + 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)))) - ((u + single(-1.0)) / (single(1.0) + exp((single(pi) / s))))))));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 2: 29.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-4 \cdot \left(\left(--1\right) - \left(-1 + s \cdot \left(\pi \cdot \mathsf{fma}\left(u, 0.5, -0.25\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* s (* -4.0 (- (- -1.0) (+ -1.0 (* s (* PI (fma u 0.5 -0.25))))))))

C

float code(float u, float s) {
	return s * (-4.0f * (-(-1.0f) - (-1.0f + (s * (((float) M_PI) * fmaf(u, 0.5f, -0.25f))))));
}

Julia

function code(u, s)
	return Float32(s * Float32(Float32(-4.0) * Float32(Float32(-Float32(-1.0)) - Float32(Float32(-1.0) + Float32(s * Float32(Float32(pi) * fma(u, Float32(0.5), Float32(-0.25))))))))
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 10.7%

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

   1. associate--r+ 10.7%

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

   2. cancel-sign-sub-inv 10.7%

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

   3. distribute-rgt-out-- 10.7%

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

   4. *-commutative 10.7%

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

   5. metadata-eval 10.7%

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

   6. metadata-eval 10.7%

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

   7. *-commutative 10.7%

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

6. Simplified 10.7%

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

   1. expm1-log1p-u 0.4%

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

   2. *-commutative 0.4%

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

   3. *-commutative 0.4%

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

   4. fma-def 0.4%

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

8. Applied egg-rr 0.4%

   \[\leadsto s \cdot \left(--4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.5, u \cdot \pi, \pi \cdot -0.25\right)}{s}\right)\right)}\right) \]
9. Step-by-step derivation

   1. fma-udef 0.4%

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

   2. *-commutative 0.4%

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

10. Applied egg-rr 0.4%

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

12. Applied egg-rr 28.8%

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

13. Final simplification 28.8%

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

Alternative 3: 29.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-4 \cdot \left(\left(--2\right) - s \cdot \left(\pi \cdot \mathsf{fma}\left(u, 0.5, -0.25\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* s (* -4.0 (- (- -2.0) (* s (* PI (fma u 0.5 -0.25)))))))

C

float code(float u, float s) {
	return s * (-4.0f * (-(-2.0f) - (s * (((float) M_PI) * fmaf(u, 0.5f, -0.25f)))));
}

Julia

function code(u, s)
	return Float32(s * Float32(Float32(-4.0) * Float32(Float32(-Float32(-2.0)) - Float32(s * Float32(Float32(pi) * fma(u, Float32(0.5), Float32(-0.25)))))))
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 10.7%

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

   1. associate--r+ 10.7%

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

   2. cancel-sign-sub-inv 10.7%

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

   3. distribute-rgt-out-- 10.7%

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

   4. *-commutative 10.7%

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

   5. metadata-eval 10.7%

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

   6. metadata-eval 10.7%

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

   7. *-commutative 10.7%

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

6. Simplified 10.7%

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

   1. expm1-log1p-u 0.4%

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

   2. *-commutative 0.4%

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

   3. *-commutative 0.4%

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

   4. fma-def 0.4%

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

8. Applied egg-rr 0.4%

   \[\leadsto s \cdot \left(--4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.5, u \cdot \pi, \pi \cdot -0.25\right)}{s}\right)\right)}\right) \]
9. Step-by-step derivation

   1. fma-udef 0.4%

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

   2. *-commutative 0.4%

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

10. Applied egg-rr 0.4%

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

12. Applied egg-rr 28.8%

    \[\leadsto s \cdot \left(--4 \cdot \color{blue}{\left(\left(s \cdot \left(\pi \cdot \mathsf{fma}\left(u, 0.5, -0.25\right)\right) + -1\right) + -1\right)}\right) \]
13. Step-by-step derivation

    1. associate-+l+ 28.8%

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

    2. metadata-eval 28.8%

       \[\leadsto s \cdot \left(--4 \cdot \left(s \cdot \left(\pi \cdot \mathsf{fma}\left(u, 0.5, -0.25\right)\right) + \color{blue}{-2}\right)\right) \]

14. Simplified 28.8%

    \[\leadsto s \cdot \left(--4 \cdot \color{blue}{\left(s \cdot \left(\pi \cdot \mathsf{fma}\left(u, 0.5, -0.25\right)\right) + -2\right)}\right) \]

15. Final simplification 28.8%

    \[\leadsto s \cdot \left(-4 \cdot \left(\left(--2\right) - s \cdot \left(\pi \cdot \mathsf{fma}\left(u, 0.5, -0.25\right)\right)\right)\right) \]

Alternative 4: 13.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(s * Float32(Float32(-4.0) * Float32(Float32(-s) / Float32(Float32(pi) * fma(u, Float32(0.5), Float32(-0.25))))))
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 10.7%

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

   1. associate--r+ 10.7%

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

   2. cancel-sign-sub-inv 10.7%

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

   3. distribute-rgt-out-- 10.7%

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

   4. *-commutative 10.7%

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

   5. metadata-eval 10.7%

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

   6. metadata-eval 10.7%

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

   7. *-commutative 10.7%

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

6. Simplified 10.7%

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

   1. expm1-log1p-u 0.4%

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

   2. *-commutative 0.4%

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

   3. *-commutative 0.4%

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

   4. fma-def 0.4%

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

8. Applied egg-rr 0.4%

   \[\leadsto s \cdot \left(--4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.5, u \cdot \pi, \pi \cdot -0.25\right)}{s}\right)\right)}\right) \]
9. Step-by-step derivation

   1. fma-udef 0.4%

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

   2. *-commutative 0.4%

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

10. Applied egg-rr 0.4%

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

12. Applied egg-rr 13.0%

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

13. Final simplification 13.0%

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

Alternative 5: 13.3% accurate, 7.0× speedup

Math

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

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(-Float32(s * Float32(pi))))
end

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u around 0 10.6%

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

6. Applied egg-rr 12.8%

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

   1. neg-sub0 12.8%

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

   2. distribute-rgt-neg-in 12.8%

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

   3. distribute-rgt-neg-in 12.8%

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

8. Simplified 12.8%

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

9. Final simplification 12.8%

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

Alternative 6: 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 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u around 0 10.6%

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

   1. neg-mul-1 10.6%

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

6. Simplified 10.6%

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

7. Final simplification 10.6%

   \[\leadsto -\pi \]

Alternative 7: 9.5% accurate, 121.7× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-s \cdot -4\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* s (- (* s -4.0))))

C

float code(float u, float s) {
	return s * -(s * -4.0f);
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * -(s * (-4.0e0))
end function

Julia

function code(u, s)
	return Float32(s * Float32(-Float32(s * Float32(-4.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -(s * single(-4.0));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 10.7%

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

   1. associate--r+ 10.7%

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

   2. cancel-sign-sub-inv 10.7%

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

   3. distribute-rgt-out-- 10.7%

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

   4. *-commutative 10.7%

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

   5. metadata-eval 10.7%

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

   6. metadata-eval 10.7%

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

   7. *-commutative 10.7%

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

6. Simplified 10.7%

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

   1. expm1-log1p-u 0.4%

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

   2. *-commutative 0.4%

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

   3. *-commutative 0.4%

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

   4. fma-def 0.4%

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

8. Applied egg-rr 0.4%

   \[\leadsto s \cdot \left(--4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.5, u \cdot \pi, \pi \cdot -0.25\right)}{s}\right)\right)}\right) \]
9. Step-by-step derivation

   1. fma-udef 0.4%

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

   2. *-commutative 0.4%

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

10. Applied egg-rr 0.4%

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

12. Applied egg-rr 9.4%

    \[\leadsto s \cdot \left(--4 \cdot \color{blue}{\frac{s \cdot \left(\pi \cdot \mathsf{fma}\left(u, 0.5, -0.25\right)\right)}{\pi \cdot \mathsf{fma}\left(u, 0.5, -0.25\right)}}\right) \]
13. Step-by-step derivation

    1. associate-/l* 9.4%

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

    2. *-inverses 9.4%

       \[\leadsto s \cdot \left(--4 \cdot \frac{s}{\color{blue}{1}}\right) \]

    3. /-rgt-identity 9.4%

       \[\leadsto s \cdot \left(--4 \cdot \color{blue}{s}\right) \]

14. Simplified 9.4%

    \[\leadsto s \cdot \left(--4 \cdot \color{blue}{s}\right) \]

15. Final simplification 9.4%

    \[\leadsto s \cdot \left(-s \cdot -4\right) \]

Reproduce

herbie shell --seed 2023312 
(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))))