Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 14.8s

Alternatives: 11

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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
      1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)));
end

Alternative 1: 98.9% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (/ PI (- s)))))
      (/ (- 1.0 u) (+ 1.0 (exp (/ 1.0 (/ s PI)))))))
    -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((1.0f / (s / ((float) M_PI)))))))) + -1.0f));
}

Julia

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

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

   1. clear-num 99.0%

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

   2. inv-pow 99.0%

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

6. Applied egg-rr 99.0%

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

   1. unpow-1 99.0%

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

8. Simplified 99.0%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     -1.0
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (- 1.0 u) (+ 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 - 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(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(Float32(1.0) - u) / 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) - u) / (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%

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

5. Final simplification 99.0%

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

Alternative 3: 24.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(fma(Float32(4.0), Float32(Float32(Float32(Float32(pi) * Float32(Float32(u * Float32(-0.25)) + Float32(0.25))) + Float32(u * Float32(Float32(pi) * Float32(-0.25)))) / s), Float32(1.0))))
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%

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

5. Taylor expanded in s around -inf 24.8%

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

   1. associate-*r/ 24.8%

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

   2. +-commutative 24.8%

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

   3. associate-*r/ 24.8%

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

   4. fma-define 24.8%

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

7. Simplified 24.8%

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

8. Final simplification 24.8%

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

Alternative 4: 24.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(\pi \cdot -0.25 + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    1.0
    (* 4.0 (/ (- (* -0.25 (* u PI)) (+ (* PI -0.25) (* 0.25 (* u PI)))) s))))))

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(1.0) + (single(4.0) * (((single(-0.25) * (u * single(pi))) - ((single(pi) * single(-0.25)) + (single(0.25) * (u * 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%

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

5. Taylor expanded in s around -inf 24.8%

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

6. Final simplification 24.8%

   \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(\pi \cdot -0.25 + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right) \]
7. Add Preprocessing

Alternative 5: 14.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{{s}^{2}}{s} \cdot \left(\frac{\pi \cdot -0.25 + \left(u \cdot \pi\right) \cdot 0.5}{s} \cdot \left(--4\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (/ (pow s 2.0) s) (* (/ (+ (* PI -0.25) (* (* u PI) 0.5)) s) (- -4.0))))

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = ((s ^ single(2.0)) / s) * ((((single(pi) * single(-0.25)) + ((u * single(pi)) * single(0.5))) / 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%

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

5. Taylor expanded in s around inf 10.9%

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

   1. associate--r+ 10.9%

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

   2. cancel-sign-sub-inv 10.9%

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

   3. distribute-rgt-out-- 10.9%

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

   4. *-commutative 10.9%

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

   5. metadata-eval 10.9%

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

   6. metadata-eval 10.9%

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

   7. *-commutative 10.9%

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

7. Simplified 10.9%

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

   1. neg-sub0 10.9%

      \[\leadsto \color{blue}{\left(0 - s\right)} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   2. flip-- 13.8%

      \[\leadsto \color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   3. metadata-eval 13.8%

      \[\leadsto \frac{\color{blue}{0} - s \cdot s}{0 + s} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   4. pow2 13.8%

      \[\leadsto \frac{0 - \color{blue}{{s}^{2}}}{0 + s} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   5. add-sqr-sqrt 13.8%

      \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   6. sqrt-unprod 7.4%

      \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s \cdot s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   7. sqr-neg 7.4%

      \[\leadsto \frac{0 - {s}^{2}}{0 + \sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   8. sqrt-unprod -0.0%

      \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{-s} \cdot \sqrt{-s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   9. add-sqr-sqrt 8.5%

      \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\left(-s\right)}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   10. sub-neg 8.5%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{0 - s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   11. neg-sub0 8.5%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{-s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   12. add-sqr-sqrt -0.0%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   13. sqrt-unprod 7.4%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   14. sqr-neg 7.4%

       \[\leadsto \frac{0 - {s}^{2}}{\sqrt{\color{blue}{s \cdot s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   15. sqrt-unprod 13.8%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

   16. add-sqr-sqrt 13.8%

       \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]

9. Applied egg-rr 13.8%

   \[\leadsto \color{blue}{\frac{0 - {s}^{2}}{s}} \cdot \left(-4 \cdot \frac{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}{s}\right) \]
10. Step-by-step derivation

    1. sub0-neg 13.8%

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

11. Simplified 13.8%

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

12. Final simplification 13.8%

    \[\leadsto \frac{{s}^{2}}{s} \cdot \left(\frac{\pi \cdot -0.25 + \left(u \cdot \pi\right) \cdot 0.5}{s} \cdot \left(--4\right)\right) \]
13. Add Preprocessing

Alternative 6: 11.7% accurate, 25.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

5. Taylor expanded in s around inf 10.9%

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

6. Final simplification 10.9%

   \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + \pi \cdot 0.25\right)\right) \]
7. Add Preprocessing

Alternative 7: 11.7% accurate, 28.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

5. Taylor expanded in s around -inf 10.9%

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

   1. associate--r+ 10.9%

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

   2. cancel-sign-sub-inv 10.9%

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

   3. metadata-eval 10.9%

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

   4. cancel-sign-sub-inv 10.9%

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

   5. associate-*r* 10.9%

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

   6. distribute-rgt-out 10.9%

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

   7. metadata-eval 10.9%

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

   8. *-commutative 10.9%

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

   9. associate-*r* 10.9%

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

   10. *-commutative 10.9%

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

7. Simplified 10.9%

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

8. Final simplification 10.9%

   \[\leadsto \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot -4 \]
9. Add Preprocessing

Alternative 8: 14.2% accurate, 39.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{\pi}{-s}\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 5.000000229068525e-19) 0.0 (* s (/ PI (- s)))))

C

float code(float u, float s) {
	float tmp;
	if (s <= 5.000000229068525e-19f) {
		tmp = 0.0f;
	} else {
		tmp = s * (((float) M_PI) / -s);
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(5.000000229068525e-19))
		tmp = Float32(0.0);
	else
		tmp = Float32(s * Float32(Float32(pi) / Float32(-s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	tmp = single(0.0);
	if (s <= single(5.000000229068525e-19))
		tmp = single(0.0);
	else
		tmp = s * (single(pi) / -s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 5.00000023e-19

   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%

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

   5. Taylor expanded in s around inf 13.0%

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

   6. Taylor expanded in s around 0 13.0%

      \[\leadsto \color{blue}{0} \]

   if 5.00000023e-19 < s

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in u around 0 15.0%

      \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\pi}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{\pi}{-s}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 14.2% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-\pi\\ \end{array} \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (if (<= s 5.000000229068525e-19) 0.0 (- PI)))

C

float code(float u, float s) {
	float tmp;
	if (s <= 5.000000229068525e-19f) {
		tmp = 0.0f;
	} else {
		tmp = -((float) M_PI);
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(5.000000229068525e-19))
		tmp = Float32(0.0);
	else
		tmp = Float32(-Float32(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	tmp = single(0.0);
	if (s <= single(5.000000229068525e-19))
		tmp = single(0.0);
	else
		tmp = -single(pi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 5.00000023e-19

   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%

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

   5. Taylor expanded in s around inf 13.0%

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

   6. Taylor expanded in s around 0 13.0%

      \[\leadsto \color{blue}{0} \]

   if 5.00000023e-19 < s

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in u around 0 15.0%

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

      1. neg-mul-1 15.0%

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

   7. Simplified 15.0%

      \[\leadsto \color{blue}{-\pi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 11.7% accurate, 61.9× 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 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%

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

5. Taylor expanded in s around inf 10.9%

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

   1. associate--r+ 10.9%

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

   2. cancel-sign-sub-inv 10.9%

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

   3. distribute-rgt-out-- 10.9%

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

   4. *-commutative 10.9%

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

   5. metadata-eval 10.9%

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

   6. metadata-eval 10.9%

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

   7. *-commutative 10.9%

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

7. Simplified 10.9%

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

8. Taylor expanded in u around 0 10.9%

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

   1. neg-mul-1 10.9%

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

   2. +-commutative 10.9%

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

   3. associate-*r* 10.9%

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

   4. neg-mul-1 10.9%

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

   5. distribute-rgt-out 10.9%

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

10. Simplified 10.9%

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

11. Final simplification 10.9%

    \[\leadsto \pi \cdot \left(-1 + u \cdot 2\right) \]
12. Add Preprocessing

Alternative 11: 10.4% accurate, 433.0× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 0.0)

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(0.0);
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%

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

5. Taylor expanded in s around inf 10.7%

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

6. Taylor expanded in s around 0 10.7%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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))))