Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 21.0s

Alternatives: 8

Speedup: 1.3×

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\pi}\right)}^{2} \cdot \frac{\sqrt[3]{\pi}}{s}}}} + -1\right) \end{array} \]

FPCore

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

C

float code(float u, float s) {
	return -s * logf(((1.0f / ((u / (1.0f + expf((((float) M_PI) / -s)))) + ((1.0f - u) / (1.0f + expf((powf(cbrtf(((float) M_PI)), 2.0f) * (cbrtf(((float) M_PI)) / s))))))) + -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((cbrt(Float32(pi)) ^ Float32(2.0)) * Float32(cbrt(Float32(pi)) / 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. Step-by-step derivation

   1. add-cube-cbrt 99.0%

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

   2. associate-/l* 99.0%

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

   3. pow2 99.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{s}}}} + -1\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2} \cdot \frac{\sqrt[3]{\pi}}{s}}}}} + -1\right) \]

7. Final simplification 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\pi}\right)}^{2} \cdot \frac{\sqrt[3]{\pi}}{s}}}} + -1\right) \]
8. Add Preprocessing

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

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(Float32(-s) * 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%

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

Alternative 3: 13.2% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 (log (pow (exp (- s)) (/ PI s))))

C

float code(float u, float s) {
	return logf(powf(expf(-s), (((float) M_PI) / s)));
}

Julia

function code(u, s)
	return log((exp(Float32(-s)) ^ Float32(Float32(pi) / s)))
end

MATLAB

function tmp = code(u, s)
	tmp = log((exp(-s) ^ (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 u around 0 11.3%

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

   1. add-log-exp 11.3%

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

   2. exp-prod 13.0%

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

7. Applied egg-rr 13.0%

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

8. Final simplification 13.0%

   \[\leadsto \log \left({\left(e^{-s}\right)}^{\left(\frac{\pi}{s}\right)}\right) \]
9. Add Preprocessing

Alternative 4: 11.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(4.0) * expm1(log1p(Float32(Float32(pi) * Float32(Float32(-0.25) + Float32(u * Float32(0.5)))))))
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 11.5%

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

   1. associate--r+ 11.5%

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

   2. cancel-sign-sub-inv 11.5%

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

   3. distribute-rgt-out-- 11.5%

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

   4. *-commutative 11.5%

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

   5. metadata-eval 11.5%

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

   6. metadata-eval 11.5%

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

   7. *-commutative 11.5%

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

7. Simplified 11.5%

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

   1. expm1-log1p-u 11.5%

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

   2. expm1-undefine 11.5%

      \[\leadsto 4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25\right)} - 1\right)} \]

   3. associate-*l* 11.5%

      \[\leadsto 4 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\pi \cdot \left(u \cdot 0.5\right)} + \pi \cdot -0.25\right)} - 1\right) \]

   4. fma-define 11.5%

      \[\leadsto 4 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\pi, u \cdot 0.5, \pi \cdot -0.25\right)}\right)} - 1\right) \]

9. Applied egg-rr 11.5%

   \[\leadsto 4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(\pi, u \cdot 0.5, \pi \cdot -0.25\right)\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-define 11.5%

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

    2. fma-undefine 11.5%

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

    3. *-commutative 11.5%

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

    4. *-commutative 11.5%

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

    5. *-commutative 11.5%

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

    6. associate-*r* 11.5%

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

    7. +-commutative 11.5%

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

    8. associate-*r* 11.5%

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

    9. *-commutative 11.5%

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

    10. distribute-rgt-out 11.5%

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

11. Simplified 11.5%

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

12. Final simplification 11.5%

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

Alternative 5: 11.7% accurate, 33.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (s * single(4.0)) * ((single(pi) * (single(-0.25) + (u * single(0.5)))) / 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 11.5%

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

   1. associate--r+ 11.5%

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

   2. cancel-sign-sub-inv 11.5%

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

   3. distribute-rgt-out-- 11.5%

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

   4. *-commutative 11.5%

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

   5. metadata-eval 11.5%

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

   6. metadata-eval 11.5%

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

   7. *-commutative 11.5%

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

7. Simplified 11.5%

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

   1. pow1 11.5%

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

   2. associate-*r* 11.5%

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

   3. associate-*l* 11.5%

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

   4. fma-define 11.5%

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

9. Applied egg-rr 11.5%

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

    1. unpow1 11.5%

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

    2. distribute-lft-neg-out 11.5%

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

    3. distribute-rgt-neg-in 11.5%

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

    4. metadata-eval 11.5%

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

    5. fma-undefine 11.5%

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

    6. *-commutative 11.5%

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

    7. *-commutative 11.5%

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

    8. *-commutative 11.5%

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

    9. associate-*r* 11.5%

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

    10. +-commutative 11.5%

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

    11. associate-*r* 11.5%

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

    12. *-commutative 11.5%

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

    13. distribute-rgt-out 11.5%

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

11. Simplified 11.5%

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

12. Final simplification 11.5%

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

Alternative 6: 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. Step-by-step derivation

   1. add-cube-cbrt 99.0%

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

   2. associate-/l* 99.0%

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

   3. pow2 99.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \frac{\sqrt[3]{\pi}}{s}}}} + -1\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2} \cdot \frac{\sqrt[3]{\pi}}{s}}}}} + -1\right) \]

7. Taylor expanded in s around inf 11.5%

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

   1. associate--r+ 11.5%

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

   2. cancel-sign-sub-inv 11.5%

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

   3. *-commutative 11.5%

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

   4. *-commutative 11.5%

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

   5. *-commutative 11.5%

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

   6. *-commutative 11.5%

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

   7. distribute-lft-out-- 11.5%

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

   8. metadata-eval 11.5%

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

   9. associate-*r* 11.5%

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

   10. *-commutative 11.5%

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

   11. metadata-eval 11.5%

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

   12. *-commutative 11.5%

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

   13. associate-*r* 11.5%

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

   14. +-commutative 11.5%

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

9. Simplified 11.5%

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

10. Final simplification 11.5%

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

Alternative 7: 11.5% accurate, 72.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (s * -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 u around 0 11.3%

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

   1. associate-*r/ 11.3%

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

7. Applied egg-rr 11.3%

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

8. Final simplification 11.3%

   \[\leadsto \frac{s \cdot \left(-\pi\right)}{s} \]
9. Add Preprocessing

Alternative 8: 11.5% accurate, 216.5× 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%

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

5. Taylor expanded in u around 0 11.3%

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

   1. neg-mul-1 11.3%

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

7. Simplified 11.3%

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

8. Final simplification 11.3%

   \[\leadsto -\pi \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(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))))