Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 12.5s

Alternatives: 10

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

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

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: 98.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return s * -logf((-1.0f + (1.0f / ((u / (1.0f + expf((((float) M_PI) / -s)))) - (-1.0f / (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(-1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(-1.0) + (single(1.0) / ((u / (single(1.0) + exp((single(pi) / -s)))) - (single(-1.0) / (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. Taylor expanded in u around 0 98.8%

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

5. Final simplification 98.8%

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

Alternative 3: 37.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{s \cdot \left(-\log \left(-1 + \frac{1}{u \cdot 0.5 - \frac{-1}{1 + e^{\frac{\pi}{s}}}}\right)\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (pow
  (cbrt
   (*
    s
    (- (log (+ -1.0 (/ 1.0 (- (* u 0.5) (/ -1.0 (+ 1.0 (exp (/ PI s)))))))))))
  3.0))

C

float code(float u, float s) {
	return powf(cbrtf((s * -logf((-1.0f + (1.0f / ((u * 0.5f) - (-1.0f / (1.0f + expf((((float) M_PI) / s)))))))))), 3.0f);
}

Julia

function code(u, s)
	return cbrt(Float32(s * Float32(-log(Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(Float32(u * Float32(0.5)) - Float32(Float32(-1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))))))))) ^ Float32(3.0)
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 u around 0 98.8%

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

5. Taylor expanded in s around inf 37.2%

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

   1. add-cube-cbrt 37.2%

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

   2. pow3 37.2%

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

7. Applied egg-rr 37.2%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{s \cdot \left(-\log \left(-1 + \frac{1}{u \cdot 0.5 - \frac{-1}{1 + e^{\frac{\pi}{s}}}}\right)\right)}\right)}^{3}} \]

8. Final simplification 37.2%

   \[\leadsto {\left(\sqrt[3]{s \cdot \left(-\log \left(-1 + \frac{1}{u \cdot 0.5 - \frac{-1}{1 + e^{\frac{\pi}{s}}}}\right)\right)}\right)}^{3} \]

Alternative 4: 37.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(-1 + \frac{1}{u \cdot 0.5 - \frac{-1}{1 + e^{\frac{\pi}{s}}}}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (- (log (+ -1.0 (/ 1.0 (- (* u 0.5) (/ -1.0 (+ 1.0 (exp (/ PI s)))))))))))

C

float code(float u, float s) {
	return s * -logf((-1.0f + (1.0f / ((u * 0.5f) - (-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(0.5)) - Float32(Float32(-1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(-1.0) + (single(1.0) / ((u * single(0.5)) - (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. Taylor expanded in u around 0 98.8%

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

5. Taylor expanded in s around inf 37.2%

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

   1. distribute-rgt-neg-out 37.2%

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

   2. +-commutative 37.2%

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

   3. div-inv 37.2%

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

   4. metadata-eval 37.2%

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

   5. metadata-eval 37.2%

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

7. Applied egg-rr 37.2%

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

8. Final simplification 37.2%

   \[\leadsto s \cdot \left(-\log \left(-1 + \frac{1}{u \cdot 0.5 - \frac{-1}{1 + e^{\frac{\pi}{s}}}}\right)\right) \]

Alternative 5: 36.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return s * -logf((-1.0f + (1.0f / ((u / 2.0f) - (-1.0f / (1.0f + (1.0f + (((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(2.0)) - Float32(Float32(-1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(pi) / s))))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(-1.0) + (single(1.0) / ((u / single(2.0)) - (single(-1.0) / (single(1.0) + (single(1.0) + (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. Taylor expanded in u around 0 98.8%

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

5. Taylor expanded in s around inf 37.2%

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

6. Taylor expanded in s around inf 35.1%

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

7. Final simplification 35.1%

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

Alternative 6: 11.6% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(-4.0) * ((single(pi) * ((u * single(-0.25)) + single(0.25))) + (u * (single(pi) * single(-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 12.2%

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

   1. associate--r+ 12.2%

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

   2. cancel-sign-sub-inv 12.2%

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

   3. cancel-sign-sub-inv 12.2%

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

   4. metadata-eval 12.2%

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

   5. associate-*r* 12.2%

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

   6. distribute-rgt-out 12.2%

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

   7. *-commutative 12.2%

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

   8. metadata-eval 12.2%

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

   9. *-commutative 12.2%

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

   10. associate-*l* 12.2%

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

6. Simplified 12.2%

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

7. Final simplification 12.2%

   \[\leadsto -4 \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \]

Alternative 7: 11.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(4.0) * Float32(Float32(pi) * Float32(Float32(u * Float32(0.25)) - fma(u, Float32(-0.25), 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 12.2%

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

   1. add-cube-cbrt 12.2%

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

   2. pow3 12.2%

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

   3. associate-*r* 12.2%

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

   4. associate-*r* 12.2%

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

   5. *-commutative 12.2%

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

   6. distribute-rgt-in 12.2%

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

   7. fma-def 12.2%

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

6. Applied egg-rr 12.2%

   \[\leadsto 4 \cdot \color{blue}{{\left(\sqrt[3]{\left(0.25 \cdot u\right) \cdot \pi - \pi \cdot \mathsf{fma}\left(u, -0.25, 0.25\right)}\right)}^{3}} \]
7. Step-by-step derivation

   1. rem-cube-cbrt 12.2%

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

   2. sub-neg 12.2%

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

   3. *-commutative 12.2%

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

   4. *-commutative 12.2%

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

8. Applied egg-rr 12.2%

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

   1. sub-neg 12.2%

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

   2. distribute-lft-out-- 12.2%

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

10. Simplified 12.2%

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

11. Final simplification 12.2%

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

Alternative 8: 11.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return 4.0f * ((((float) M_PI) * -0.25f) + (0.5f * (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) * Float32(u * Float32(pi)))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * ((single(pi) * single(-0.25)) + (single(0.5) * (u * 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 s around inf 12.2%

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

   1. associate--r+ 12.2%

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

   2. cancel-sign-sub-inv 12.2%

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

   3. distribute-rgt-out-- 12.2%

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

   4. *-commutative 12.2%

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

   5. metadata-eval 12.2%

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

   6. metadata-eval 12.2%

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

   7. *-commutative 12.2%

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

6. Simplified 12.2%

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

7. Final simplification 12.2%

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

Alternative 9: 11.6% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * (single(pi) * ((u * single(0.5)) + single(-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. Step-by-step derivation

   1. clear-num 99.0%

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

   2. associate-/r/ 99.0%

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

5. Applied egg-rr 99.0%

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

6. Taylor expanded in s around inf 12.2%

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

   1. associate--r+ 12.2%

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

   2. cancel-sign-sub-inv 12.2%

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

   3. *-commutative 12.2%

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

   4. *-commutative 12.2%

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

   5. distribute-rgt-out-- 12.2%

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

   6. metadata-eval 12.2%

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

   7. *-commutative 12.2%

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

   8. *-commutative 12.2%

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

   9. metadata-eval 12.2%

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

   10. +-commutative 12.2%

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

   11. *-commutative 12.2%

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

   12. *-commutative 12.2%

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

   13. associate-*r* 12.2%

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

   14. *-commutative 12.2%

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

   15. distribute-rgt-out 12.2%

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

8. Simplified 12.2%

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

9. Final simplification 12.2%

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

Alternative 10: 11.4% 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 11.9%

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

   1. neg-mul-1 11.9%

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

6. Simplified 11.9%

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

7. Final simplification 11.9%

   \[\leadsto -\pi \]

Reproduce

herbie shell --seed 2023308 
(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))))