Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 17.5s

Alternatives: 11

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, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (- (/ PI s)))))
      (/ (- 1.0 u) (+ 1.0 (pow E (/ 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 + powf(((float) M_E), (((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(Float32(pi) / s))))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(Float32(pi) / s)))))) + Float32(-1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / ((u / (single(1.0) + exp(-(single(pi) / s)))) + ((single(1.0) - u) / (single(1.0) + (single(2.71828182845904523536) ^ (single(pi) / s)))))) + single(-1.0)));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \left(e^{1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}\right)\right)\right)\right)\right), -1\right)\right)\right) \]

   2. exp-prod N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \left({\left(e^{1}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}\right)\right)\right)\right)\right), -1\right)\right)\right) \]

   3. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\left(e^{1}\right), \left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right)\right)\right)\right)\right), -1\right)\right)\right) \]

   4. exp-1-e N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right)\right)\right)\right)\right), -1\right)\right)\right) \]

   5. E-lowering-E.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right)\right)\right)\right)\right), -1\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(\mathsf{PI}\left(\right), s\right)\right)\right)\right)\right)\right), -1\right)\right)\right) \]

   7. PI-lowering-PI.f32 98.8%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right)\right), -1\right)\right)\right) \]

6. Applied egg-rr 98.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + \color{blue}{{e}^{\left(\frac{\pi}{s}\right)}}}} + -1\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\mathsf{PI}\left(\right)}{s}}} + \frac{1 - u}{1 + {\mathsf{E}\left(\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + -1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \]

   2. distribute-rgt-neg-out N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\mathsf{PI}\left(\right)}{s}}} + \frac{1 - u}{1 + {\mathsf{E}\left(\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + -1\right) \cdot s\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \left(\mathsf{neg}\left(\log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\mathsf{PI}\left(\right)}{s}}} + \frac{1 - u}{1 + {\mathsf{E}\left(\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + -1\right)\right)\right) \cdot \color{blue}{s} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\mathsf{PI}\left(\right)}{s}}} + \frac{1 - u}{1 + {\mathsf{E}\left(\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + -1\right)\right)\right), \color{blue}{s}\right) \]

8. Applied egg-rr 98.8%

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

   1. *-lft-identity N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\left(e^{1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}\right), 1\right)\right)\right)\right), -1\right)\right)\right), s\right) \]

   2. exp-prod N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\left({\left(e^{1}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}\right), 1\right)\right)\right)\right), -1\right)\right)\right), s\right) \]

   3. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\left(e^{1}\right), \left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right), 1\right)\right)\right)\right), -1\right)\right)\right), s\right) \]

   4. exp-1-e N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{E}\left(\right), \left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right), 1\right)\right)\right)\right), -1\right)\right)\right), s\right) \]

   5. E-lowering-E.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right), 1\right)\right)\right)\right), -1\right)\right)\right), s\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(\mathsf{PI}\left(\right), s\right)\right), 1\right)\right)\right)\right), -1\right)\right)\right), s\right) \]

   7. PI-lowering-PI.f32 98.8%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(\mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right), 1\right)\right)\right)\right), -1\right)\right)\right), s\right) \]

10. Applied egg-rr 98.8%

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

11. Final simplification 98.8%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + {e}^{\left(\frac{\pi}{s}\right)}}} + -1\right) \]
12. 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(Float32(pi) / 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 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\mathsf{PI}\left(\right)}{s}}} + \frac{1 - u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} + -1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \]

   2. neg-mul-1 N/A

      \[\leadsto \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\mathsf{PI}\left(\right)}{s}}} + \frac{1 - u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} + -1\right) \cdot \left(-1 \cdot \color{blue}{s}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\mathsf{PI}\left(\right)}{s}}} + \frac{1 - u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} + -1\right) \cdot -1\right) \cdot \color{blue}{s} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\mathsf{PI}\left(\right)}{s}}} + \frac{1 - u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} + -1\right) \cdot -1\right), \color{blue}{s}\right) \]

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 3: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around -inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}\right)\right)\right)\right), -1\right)\right)\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \left(1 + \left(\mathsf{neg}\left(\frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)\right)\right)\right)\right)\right)\right), -1\right)\right)\right) \]

   2. unsub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)\right)\right)\right)\right), -1\right)\right)\right) \]

   3. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)\right)\right)\right)\right)\right), -1\right)\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}\right), s\right)\right)\right)\right)\right)\right), -1\right)\right)\right) \]

7. Simplified 94.2%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + \color{blue}{\left(1 - \frac{\left(-\pi\right) - \frac{\frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.16666666666666666}{s} + \pi \cdot \left(\pi \cdot 0.5\right)}{s}}{s}\right)}}} + -1\right) \]

8. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(-1 \cdot s\right), \color{blue}{\log \left(\left(\frac{1}{u} + \frac{e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}{u}\right) - 1\right)}\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{neg}\left(s\right)\right), \log \color{blue}{\left(\left(\frac{1}{u} + \frac{e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}{u}\right) - 1\right)}\right) \]

   4. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \log \color{blue}{\left(\left(\frac{1}{u} + \frac{e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}{u}\right) - 1\right)}\right) \]

   5. log-lowering-log.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\left(\left(\frac{1}{u} + \frac{e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}{u}\right) - 1\right)\right)\right) \]

   6. associate--l+ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\left(\frac{1}{u} + \left(\frac{e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}{u} - 1\right)\right)\right)\right) \]

   7. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{u}\right), \left(\frac{e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}{u} - 1\right)\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u\right), \left(\frac{e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}{u} - 1\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u\right), \left(\frac{e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}{u} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u\right), \left(\frac{e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}{u} + -1\right)\right)\right)\right) \]

   11. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u\right), \mathsf{+.f32}\left(\left(\frac{e^{\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}{u}\right), -1\right)\right)\right)\right) \]

10. Simplified 96.9%

    \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{u} + \left(\frac{e^{0 - \frac{\pi}{s}}}{u} + -1\right)\right)} \]

11. Final simplification 96.9%

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

Alternative 4: 24.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(1.0) + ((((single(pi) * (u * single(0.5))) + (single(pi) * single(-0.25))) * single(-4.0)) / s)));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \left(e^{1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}\right)\right)\right)\right)\right), -1\right)\right)\right) \]

   2. exp-prod N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \left({\left(e^{1}\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}\right)\right)\right)\right)\right), -1\right)\right)\right) \]

   3. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\left(e^{1}\right), \left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right)\right)\right)\right)\right), -1\right)\right)\right) \]

   4. exp-1-e N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right)\right)\right)\right)\right), -1\right)\right)\right) \]

   5. E-lowering-E.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right)\right)\right)\right)\right), -1\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(\mathsf{PI}\left(\right), s\right)\right)\right)\right)\right)\right), -1\right)\right)\right) \]

   7. PI-lowering-PI.f32 98.8%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \mathsf{+.f32}\left(1, \mathsf{exp.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(1, u\right), \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right)\right)\right)\right), -1\right)\right)\right) \]

6. Applied egg-rr 98.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + \color{blue}{{e}^{\left(\frac{\pi}{s}\right)}}}} + -1\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\mathsf{PI}\left(\right)}{s}}} + \frac{1 - u}{1 + {\mathsf{E}\left(\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + -1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(s\right)\right)} \]

   2. distribute-rgt-neg-out N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\mathsf{PI}\left(\right)}{s}}} + \frac{1 - u}{1 + {\mathsf{E}\left(\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + -1\right) \cdot s\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \left(\mathsf{neg}\left(\log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\mathsf{PI}\left(\right)}{s}}} + \frac{1 - u}{1 + {\mathsf{E}\left(\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + -1\right)\right)\right) \cdot \color{blue}{s} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\mathsf{PI}\left(\right)}{s}}} + \frac{1 - u}{1 + {\mathsf{E}\left(\right)}^{\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}}} + -1\right)\right)\right), \color{blue}{s}\right) \]

8. Applied egg-rr 98.8%

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

9. Taylor expanded in s around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\color{blue}{\left(1 + -4 \cdot \frac{\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right)\right), s\right) \]
10. Step-by-step derivation

    1. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \left(-4 \cdot \frac{\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)\right)\right)\right), s\right) \]

    2. associate-*r/ N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{-4 \cdot \left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{s}\right)\right)\right)\right), s\right) \]

    3. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(\mathsf{log.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(-4 \cdot \left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)\right), s\right)\right)\right)\right), s\right) \]

11. Simplified 25.5%

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

12. Final simplification 25.5%

    \[\leadsto \left(-s\right) \cdot \log \left(1 + \frac{\left(\pi \cdot \left(u \cdot 0.5\right) + \pi \cdot -0.25\right) \cdot -4}{s}\right) \]
13. Add Preprocessing

Alternative 5: 14.0% accurate, 27.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(pi) / s) * ((-s * (s + (u * (s * single(-2.0))))) / s);
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-4 \cdot \frac{\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right) \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{-4 \cdot \left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{s}}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot -4}{s}\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{-4}{s}}\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{*.f32}\left(\left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left(\frac{-4}{s}\right)}\right)\right) \]

7. Simplified 12.2%

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

8. Taylor expanded in u around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s} + \frac{\mathsf{PI}\left(\right)}{s}\right)}\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\mathsf{PI}\left(\right)}{s} + \color{blue}{-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}}\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \color{blue}{\left(-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}\right)}\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI}\left(\right), s\right), \left(\color{blue}{-2} \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}\right)\right)\right) \]

   4. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \left(-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \left(\frac{-2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{s}}\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\left(-2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{s}\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \left(u \cdot \mathsf{PI}\left(\right)\right)\right), s\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \left(\mathsf{PI}\left(\right) \cdot u\right)\right), s\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), u\right)\right), s\right)\right)\right) \]

   10. PI-lowering-PI.f32 12.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), u\right)\right), s\right)\right)\right) \]

10. Simplified 12.2%

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

    1. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{-2 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)}{s} + \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right) \]

    2. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{-2 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)}{s} + \frac{1}{\color{blue}{\frac{s}{\mathsf{PI}\left(\right)}}}\right)\right) \]

    3. frac-add N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right) \cdot \frac{s}{\mathsf{PI}\left(\right)} + s \cdot 1}{\color{blue}{s \cdot \frac{s}{\mathsf{PI}\left(\right)}}}\right)\right) \]

    4. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{/.f32}\left(\left(\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot u\right)\right) \cdot \frac{s}{\mathsf{PI}\left(\right)} + s \cdot 1\right), \color{blue}{\left(s \cdot \frac{s}{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

12. Applied egg-rr 10.7%

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

    1. associate-*r/ N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\frac{u \cdot \left(\mathsf{PI}\left(\right) \cdot -2\right)}{\frac{\mathsf{PI}\left(\right)}{s}} + s\right)}{\color{blue}{\frac{s}{\frac{\mathsf{PI}\left(\right)}{s}}}} \]

    2. associate-/r/ N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\frac{u \cdot \left(\mathsf{PI}\left(\right) \cdot -2\right)}{\frac{\mathsf{PI}\left(\right)}{s}} + s\right)}{s} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}} \]

    3. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{\left(\mathsf{neg}\left(s\right)\right) \cdot \left(\frac{u \cdot \left(\mathsf{PI}\left(\right) \cdot -2\right)}{\frac{\mathsf{PI}\left(\right)}{s}} + s\right)}{s}\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}\right) \]

14. Applied egg-rr 14.7%

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

15. Final simplification 14.7%

    \[\leadsto \frac{\pi}{s} \cdot \frac{\left(-s\right) \cdot \left(s + u \cdot \left(s \cdot -2\right)\right)}{s} \]
16. Add Preprocessing

Alternative 6: 11.5% accurate, 30.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * ((single(pi) / s) * ((single(-0.25) + (u / single(2.0))) / -single(-0.25)));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-4 \cdot \frac{\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right) \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{-4 \cdot \left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{s}}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot -4}{s}\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{-4}{s}}\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{*.f32}\left(\left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left(\frac{-4}{s}\right)}\right)\right) \]

7. Simplified 12.2%

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

   1. clear-num N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot u\right) \cdot \frac{1}{2} + \mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right) \cdot \frac{1}{\color{blue}{\frac{s}{-4}}}\right)\right) \]

   2. un-div-inv N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\left(\mathsf{PI}\left(\right) \cdot u\right) \cdot \frac{1}{2} + \mathsf{PI}\left(\right) \cdot \frac{-1}{4}}{\color{blue}{\frac{s}{-4}}}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\mathsf{PI}\left(\right) \cdot \left(u \cdot \frac{1}{2}\right) + \mathsf{PI}\left(\right) \cdot \frac{-1}{4}}{\frac{s}{-4}}\right)\right) \]

   4. distribute-lft-out N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\mathsf{PI}\left(\right) \cdot \left(u \cdot \frac{1}{2} + \frac{-1}{4}\right)}{\frac{\color{blue}{s}}{-4}}\right)\right) \]

   5. div-inv N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\mathsf{PI}\left(\right) \cdot \left(u \cdot \frac{1}{2} + \frac{-1}{4}\right)}{s \cdot \color{blue}{\frac{1}{-4}}}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\mathsf{PI}\left(\right) \cdot \left(u \cdot \frac{1}{2} + \frac{-1}{4}\right)}{s \cdot \frac{-1}{4}}\right)\right) \]

   7. times-frac N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\mathsf{PI}\left(\right)}{s} \cdot \color{blue}{\frac{u \cdot \frac{1}{2} + \frac{-1}{4}}{\frac{-1}{4}}}\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{*.f32}\left(\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \color{blue}{\left(\frac{u \cdot \frac{1}{2} + \frac{-1}{4}}{\frac{-1}{4}}\right)}\right)\right) \]

   9. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{PI}\left(\right), s\right), \left(\frac{\color{blue}{u \cdot \frac{1}{2} + \frac{-1}{4}}}{\frac{-1}{4}}\right)\right)\right) \]

   10. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \left(\frac{\color{blue}{u \cdot \frac{1}{2}} + \frac{-1}{4}}{\frac{-1}{4}}\right)\right)\right) \]

   11. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\left(u \cdot \frac{1}{2} + \frac{-1}{4}\right), \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(u \cdot \frac{1}{2}\right), \frac{-1}{4}\right), \frac{-1}{4}\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(u \cdot \frac{1}{2}\right), \frac{-1}{4}\right), \frac{-1}{4}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(u \cdot \frac{1}{1 + 1}\right), \frac{-1}{4}\right), \frac{-1}{4}\right)\right)\right) \]

   15. div-inv N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\left(\frac{u}{1 + 1}\right), \frac{-1}{4}\right), \frac{-1}{4}\right)\right)\right) \]

   16. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(u, \left(1 + 1\right)\right), \frac{-1}{4}\right), \frac{-1}{4}\right)\right)\right) \]

   17. metadata-eval 12.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(u, 2\right), \frac{-1}{4}\right), \frac{-1}{4}\right)\right)\right) \]

9. Applied egg-rr 12.2%

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

10. Final simplification 12.2%

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

Alternative 7: 11.5% accurate, 33.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s / (single(1.0) / ((single(pi) * (single(-1.0) - (u * single(-2.0)))) / s));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-4 \cdot \frac{\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right) \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{-4 \cdot \left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{s}}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot -4}{s}\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{-4}{s}}\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{*.f32}\left(\left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left(\frac{-4}{s}\right)}\right)\right) \]

7. Simplified 12.2%

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

8. Taylor expanded in u around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s} + \frac{\mathsf{PI}\left(\right)}{s}\right)}\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\mathsf{PI}\left(\right)}{s} + \color{blue}{-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}}\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \color{blue}{\left(-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}\right)}\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI}\left(\right), s\right), \left(\color{blue}{-2} \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}\right)\right)\right) \]

   4. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \left(-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \left(\frac{-2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{s}}\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\left(-2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{s}\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \left(u \cdot \mathsf{PI}\left(\right)\right)\right), s\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \left(\mathsf{PI}\left(\right) \cdot u\right)\right), s\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), u\right)\right), s\right)\right)\right) \]

   10. PI-lowering-PI.f32 12.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), u\right)\right), s\right)\right)\right) \]

10. Simplified 12.2%

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

12. Applied egg-rr 12.2%

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

13. Final simplification 12.2%

    \[\leadsto \frac{s}{\frac{1}{\frac{\pi \cdot \left(-1 - u \cdot -2\right)}{s}}} \]
14. Add Preprocessing

Alternative 8: 11.5% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * ((single(pi) * (single(-1.0) - (u * single(-2.0)))) / s);
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-4 \cdot \frac{\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right) \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{-4 \cdot \left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{s}}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot -4}{s}\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{-4}{s}}\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{*.f32}\left(\left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left(\frac{-4}{s}\right)}\right)\right) \]

7. Simplified 12.2%

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

8. Taylor expanded in u around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s} + \frac{\mathsf{PI}\left(\right)}{s}\right)}\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\mathsf{PI}\left(\right)}{s} + \color{blue}{-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}}\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \color{blue}{\left(-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}\right)}\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI}\left(\right), s\right), \left(\color{blue}{-2} \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}\right)\right)\right) \]

   4. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \left(-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \left(\frac{-2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{s}}\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\left(-2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{s}\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \left(u \cdot \mathsf{PI}\left(\right)\right)\right), s\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \left(\mathsf{PI}\left(\right) \cdot u\right)\right), s\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), u\right)\right), s\right)\right)\right) \]

   10. PI-lowering-PI.f32 12.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), u\right)\right), s\right)\right)\right) \]

10. Simplified 12.2%

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

12. Applied egg-rr 12.2%

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

13. Final simplification 12.2%

    \[\leadsto s \cdot \frac{\pi \cdot \left(-1 - u \cdot -2\right)}{s} \]
14. Add Preprocessing

Alternative 9: 11.5% 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 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-4 \cdot \frac{\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)}\right) \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{-4 \cdot \left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{s}}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot -4}{s}\right)\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{-4}{s}}\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{*.f32}\left(\left(\frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left(\frac{-4}{s}\right)}\right)\right) \]

7. Simplified 12.2%

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

8. Taylor expanded in u around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s} + \frac{\mathsf{PI}\left(\right)}{s}\right)}\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\frac{\mathsf{PI}\left(\right)}{s} + \color{blue}{-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}}\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\left(\frac{\mathsf{PI}\left(\right)}{s}\right), \color{blue}{\left(-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}\right)}\right)\right) \]

   3. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI}\left(\right), s\right), \left(\color{blue}{-2} \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}\right)\right)\right) \]

   4. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \left(-2 \cdot \frac{u \cdot \mathsf{PI}\left(\right)}{s}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \left(\frac{-2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{s}}\right)\right)\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\left(-2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{s}\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \left(u \cdot \mathsf{PI}\left(\right)\right)\right), s\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \left(\mathsf{PI}\left(\right) \cdot u\right)\right), s\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), u\right)\right), s\right)\right)\right) \]

   10. PI-lowering-PI.f32 12.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right), \mathsf{/.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), u\right)\right), s\right)\right)\right) \]

10. Simplified 12.2%

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

11. Taylor expanded in s around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. associate-*r* N/A

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

    4. metadata-eval N/A

       \[\leadsto 2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \mathsf{PI}\left(\right) \]

    5. associate-*r* N/A

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

    6. distribute-rgt-out N/A

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

    7. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{\left(2 \cdot u + -1\right)}\right) \]

    8. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \left(\color{blue}{2 \cdot u} + -1\right)\right) \]

    9. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\left(2 \cdot u\right), \color{blue}{-1}\right)\right) \]

    10. *-lowering-*.f32 12.2%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, u\right), -1\right)\right) \]

13. Simplified 12.2%

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

14. Final simplification 12.2%

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

Alternative 10: 11.3% accurate, 61.9× speedup

Math

\[\begin{array}{l} \\ 0 - s \cdot \frac{\pi}{s} \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (- 0.0 (* s (/ PI s))))

C

float code(float u, float s) {
	return 0.0f - (s * (((float) M_PI) / s));
}

Julia

function code(u, s)
	return Float32(Float32(0.0) - Float32(s * Float32(Float32(pi) / s)))
end

MATLAB

function tmp = code(u, s)
	tmp = single(0.0) - (s * (single(pi) / s));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in u around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{/.f32}\left(\mathsf{PI}\left(\right), \color{blue}{s}\right)\right) \]

   2. PI-lowering-PI.f32 12.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right) \]

7. Simplified 12.0%

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

8. Final simplification 12.0%

   \[\leadsto 0 - s \cdot \frac{\pi}{s} \]
9. Add Preprocessing

Alternative 11: 11.3% 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 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
4. Add Preprocessing

5. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \mathsf{PI}\left(\right)} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\mathsf{PI}\left(\right)\right) \]

   2. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{PI}\left(\right)\right) \]

   3. PI-lowering-PI.f32 12.0%

      \[\leadsto \mathsf{neg.f32}\left(\mathsf{PI.f32}\left(\right)\right) \]

7. Simplified 12.0%

   \[\leadsto \color{blue}{-\pi} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(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))))