Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 54.7s

Alternatives: 15

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - u}{1 + e^{\frac{\pi}{s}}} + \frac{u}{1 + e^{0 - \frac{\pi}{s}}}\\ s \cdot \log \left(\frac{{t\_0}^{-2} + \left(1 + \frac{1}{t\_0}\right)}{-1 + {t\_0}^{-3}}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0
         (+
          (/ (- 1.0 u) (+ 1.0 (exp (/ PI s))))
          (/ u (+ 1.0 (exp (- 0.0 (/ PI s))))))))
   (*
    s
    (log (/ (+ (pow t_0 -2.0) (+ 1.0 (/ 1.0 t_0))) (+ -1.0 (pow t_0 -3.0)))))))

C

float code(float u, float s) {
	float t_0 = ((1.0f - u) / (1.0f + expf((((float) M_PI) / s)))) + (u / (1.0f + expf((0.0f - (((float) M_PI) / s)))));
	return s * logf(((powf(t_0, -2.0f) + (1.0f + (1.0f / t_0))) / (-1.0f + powf(t_0, -3.0f))));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))) + Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(0.0) - Float32(Float32(pi) / s))))))
	return Float32(s * log(Float32(Float32((t_0 ^ Float32(-2.0)) + Float32(Float32(1.0) + Float32(Float32(1.0) / t_0))) / Float32(Float32(-1.0) + (t_0 ^ Float32(-3.0))))))
end

MATLAB

function tmp = code(u, s)
	t_0 = ((single(1.0) - u) / (single(1.0) + exp((single(pi) / s)))) + (u / (single(1.0) + exp((single(0.0) - (single(pi) / s)))));
	tmp = s * log((((t_0 ^ single(-2.0)) + (single(1.0) + (single(1.0) / t_0))) / (single(-1.0) + (t_0 ^ single(-3.0)))));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

6. Applied egg-rr 99.0%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(-\log \left(\frac{{\left(\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 + \frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}{-1 + {\left(\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}\right)\right)} \]

7. Final simplification 99.0%

   \[\leadsto s \cdot \log \left(\frac{{\left(\frac{1 - u}{1 + e^{\frac{\pi}{s}}} + \frac{u}{1 + e^{0 - \frac{\pi}{s}}}\right)}^{-2} + \left(1 + \frac{1}{\frac{1 - u}{1 + e^{\frac{\pi}{s}}} + \frac{u}{1 + e^{0 - \frac{\pi}{s}}}}\right)}{-1 + {\left(\frac{1 - u}{1 + e^{\frac{\pi}{s}}} + \frac{u}{1 + e^{0 - \frac{\pi}{s}}}\right)}^{-3}}\right) \]
8. Add Preprocessing

Alternative 2: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1}{\frac{1 - u}{1 + e^{\frac{\pi}{s}}} + \frac{u}{1 + e^{0 - \frac{\pi}{s}}}} + -1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    (/
     1.0
     (+
      (/ (- 1.0 u) (+ 1.0 (exp (/ PI s))))
      (/ u (+ 1.0 (exp (- 0.0 (/ PI s)))))))
    -1.0))))

C

float code(float u, float s) {
	return -s * logf(((1.0f / (((1.0f - u) / (1.0f + expf((((float) M_PI) / s)))) + (u / (1.0f + expf((0.0f - (((float) M_PI) / s))))))) + -1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))) + Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(0.0) - Float32(Float32(pi) / s))))))) + Float32(-1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / (((single(1.0) - u) / (single(1.0) + exp((single(pi) / s)))) + (u / (single(1.0) + exp((single(0.0) - (single(pi) / s))))))) + single(-1.0)));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

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

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

6. Applied egg-rr 98.9%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{1 - {\left(\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2}}{-1 - \frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right)} \]
7. Step-by-step derivation

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. inv-pow N/A

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

   5. inv-pow N/A

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 3: 97.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1 + \left(e^{0 - \frac{\pi}{s}} - u\right)}{u}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (/ (+ 1.0 (- (exp (- 0.0 (/ PI s))) u)) u))))

C

float code(float u, float s) {
	return -s * logf(((1.0f + (expf((0.0f - (((float) M_PI) / s))) - u)) / u));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) + Float32(exp(Float32(Float32(0.0) - Float32(Float32(pi) / s))) - u)) / u)))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) + (exp((single(0.0) - (single(pi) / s))) - u)) / u));
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%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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. +-lowering-+.f32 N/A

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

   2. associate-*r/ N/A

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

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

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

7. Simplified 94.0%

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

9. Applied egg-rr 94.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{\frac{\pi + \frac{\pi \cdot \left(\pi \cdot 0.5\right) - \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.16666666666666666}{s}}{s}}{s} + 2}} - 1\right)} \]

10. Taylor expanded in s around 0

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

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

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

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

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

    3. neg-mul-1 N/A

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

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

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

    5. neg-sub0 N/A

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

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

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

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

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

    8. PI-lowering-PI.f32 97.6%

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

12. Simplified 97.6%

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

13. Taylor expanded in u around 0

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

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

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

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

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

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

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

    4. neg-mul-1 N/A

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

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

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

    6. neg-mul-1 N/A

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

    7. distribute-neg-frac2 N/A

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

    8. mul-1-neg N/A

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

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

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

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

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

    11. mul-1-neg N/A

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

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

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

    13. mul-1-neg N/A

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

    14. neg-lowering-neg.f32 97.6%

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

15. Simplified 97.6%

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

16. Final simplification 97.6%

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

Alternative 4: 97.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \frac{1 + e^{0 - \frac{\pi}{s}}}{u}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (+ -1.0 (/ (+ 1.0 (exp (- 0.0 (/ PI s)))) u)))))

C

float code(float u, float s) {
	return -s * logf((-1.0f + ((1.0f + expf((0.0f - (((float) M_PI) / s)))) / u)));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(Float32(1.0) + exp(Float32(Float32(0.0) - Float32(Float32(pi) / s)))) / u))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(-1.0) + ((single(1.0) + exp((single(0.0) - (single(pi) / s)))) / u)));
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%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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. +-lowering-+.f32 N/A

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

   2. associate-*r/ N/A

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

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

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

7. Simplified 94.0%

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

9. Applied egg-rr 94.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{\frac{\pi + \frac{\pi \cdot \left(\pi \cdot 0.5\right) - \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.16666666666666666}{s}}{s}}{s} + 2}} - 1\right)} \]

10. Taylor expanded in s around 0

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

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

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

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

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

    3. neg-mul-1 N/A

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

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

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

    5. neg-sub0 N/A

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

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

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

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

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

    8. PI-lowering-PI.f32 97.6%

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

12. Simplified 97.6%

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

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

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

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

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

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

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

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

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

    5. sub0-neg N/A

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

    6. distribute-frac-neg2 N/A

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

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

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

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

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

    9. neg-lowering-neg.f32 97.6%

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

14. Applied egg-rr 97.6%

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

15. Final simplification 97.6%

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

Alternative 5: 76.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{1 + e^{0 - \frac{\pi}{s}}}{u}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (/ (+ 1.0 (exp (- 0.0 (/ PI s)))) u))))

C

float code(float u, float s) {
	return -s * logf(((1.0f + expf((0.0f - (((float) M_PI) / s)))) / u));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) + exp(Float32(Float32(0.0) - Float32(Float32(pi) / s)))) / u)))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) + exp((single(0.0) - (single(pi) / s)))) / u));
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%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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. +-lowering-+.f32 N/A

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

   2. associate-*r/ N/A

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

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

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

7. Simplified 94.0%

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

9. Applied egg-rr 94.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{\frac{\pi + \frac{\pi \cdot \left(\pi \cdot 0.5\right) - \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.16666666666666666}{s}}{s}}{s} + 2}} - 1\right)} \]

10. Taylor expanded in s around 0

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

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

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

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

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

    3. neg-mul-1 N/A

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

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

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

    5. neg-sub0 N/A

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

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

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

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

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

    8. PI-lowering-PI.f32 97.6%

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

12. Simplified 97.6%

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

13. Taylor expanded in u around 0

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

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

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

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

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

    3. neg-mul-1 N/A

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

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

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

    5. neg-mul-1 N/A

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

    6. distribute-neg-frac2 N/A

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

    7. mul-1-neg N/A

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

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

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

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

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

    10. mul-1-neg N/A

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

    11. neg-lowering-neg.f32 77.2%

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

15. Simplified 77.2%

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

16. Final simplification 77.2%

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

Alternative 6: 37.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = -s * log(((-1.0e0) + (2.0e0 / u)))
end function

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(-1.0) + (single(2.0) / u)));
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%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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. +-lowering-+.f32 N/A

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

   2. associate-*r/ N/A

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

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

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

7. Simplified 94.0%

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

9. Applied egg-rr 94.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{\frac{\pi + \frac{\pi \cdot \left(\pi \cdot 0.5\right) - \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -0.16666666666666666}{s}}{s}}{s} + 2}} - 1\right)} \]

10. Taylor expanded in s around 0

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

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

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

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

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

    3. neg-mul-1 N/A

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

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

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

    5. neg-sub0 N/A

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

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

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

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

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

    8. PI-lowering-PI.f32 97.6%

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

12. Simplified 97.6%

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

13. Taylor expanded in s around inf

    \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\color{blue}{\left(\frac{2}{u}\right)}, 1\right)\right)\right) \]
14. Step-by-step derivation

    1. /-lowering-/.f32 37.3%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(2, u\right), 1\right)\right)\right) \]

15. Simplified 37.3%

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

16. Final simplification 37.3%

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

Alternative 7: 25.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (- s) (log1p (/ PI s))))

C

float code(float u, float s) {
	return -s * log1pf((((float) M_PI) / s));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log1p(Float32(Float32(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%

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

5. Taylor expanded in s around -inf

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

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

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

7. Simplified 25.2%

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

8. Taylor expanded in u around 0

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

   1. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right)\right) \]

   2. log1p-lowering-log1p.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI}\left(\right), s\right)\right)\right) \]

   4. PI-lowering-PI.f32 25.5%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right) \]

10. Simplified 25.5%

    \[\leadsto \left(-s\right) \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]
11. Add Preprocessing

Alternative 8: 25.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log((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%

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

5. Taylor expanded in s around -inf

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

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

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

7. Simplified 25.2%

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

8. Taylor expanded in s around 0

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

   1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

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

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

   13. *-lowering-*.f32 25.2%

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

10. Simplified 25.2%

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

11. Taylor expanded in u around 0

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

    1. associate-*r* N/A

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

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

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

    3. mul-1-neg N/A

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

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

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

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

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

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

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

    7. PI-lowering-PI.f32 25.4%

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

13. Simplified 25.4%

    \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{\pi}{s}\right)} \]
14. Add Preprocessing

Alternative 9: 11.4% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf

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

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

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

   2. associate--r+ N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

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

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

   6. distribute-rgt-out-- N/A

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

   7. metadata-eval N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

   14. PI-lowering-PI.f32 11.5%

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

7. Simplified 11.5%

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

8. Final simplification 11.5%

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

Alternative 10: 13.7% accurate, 43.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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 11.3%

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

7. Simplified 11.3%

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

   1. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(0 - s\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{PI.f32}\left(\right)}, s\right)\right) \]

   2. flip-- N/A

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

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(0 \cdot 0 - s \cdot s\right), \left(0 + s\right)\right), \mathsf{/.f32}\left(\color{blue}{\mathsf{PI.f32}\left(\right)}, s\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\left(0 - s \cdot s\right), \left(0 + s\right)\right), \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \left(s \cdot s\right)\right), \left(0 + s\right)\right), \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(s, s\right)\right), \left(0 + s\right)\right), \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right) \]

   7. +-lowering-+.f32 13.9%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{+.f32}\left(0, s\right)\right), \mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right) \]

9. Applied egg-rr 13.9%

   \[\leadsto \color{blue}{\frac{0 - s \cdot s}{0 + s}} \cdot \frac{\pi}{s} \]

10. Final simplification 13.9%

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

Alternative 11: 11.2% accurate, 61.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{s}{s \cdot \pi}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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 11.3%

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

7. Simplified 11.3%

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

   1. associate-*r/ N/A

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

   2. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{s}{\left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{PI}\left(\right)}}} \]

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

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

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(s, \color{blue}{\left(\left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(s, \mathsf{*.f32}\left(\left(\mathsf{neg}\left(s\right)\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

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

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

   7. PI-lowering-PI.f32 11.3%

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

9. Applied egg-rr 11.3%

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

10. Final simplification 11.3%

    \[\leadsto \frac{-1}{\frac{s}{s \cdot \pi}} \]
11. Add Preprocessing

Alternative 12: 11.2% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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 11.3%

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

7. Simplified 11.3%

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

   1. clear-num N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \frac{1}{\color{blue}{\frac{s}{\mathsf{PI}\left(\right)}}} \]

   2. un-div-inv N/A

      \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\color{blue}{\frac{s}{\mathsf{PI}\left(\right)}}} \]

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

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

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

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

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

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

   6. PI-lowering-PI.f32 11.3%

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

9. Applied egg-rr 11.3%

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

10. Final simplification 11.3%

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

Alternative 13: 11.2% accurate, 72.2× speedup

Math

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

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(Float32(-s) * 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%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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 11.3%

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

7. Simplified 11.3%

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

   1. associate-*r/ N/A

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

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

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

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

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

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

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

   5. PI-lowering-PI.f32 11.3%

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

9. Applied egg-rr 11.3%

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

Alternative 14: 11.2% 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%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{0 - \frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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 11.3%

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

7. Simplified 11.3%

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

Alternative 15: 10.3% accurate, 433.0× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 0.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log.f32}\left(\color{blue}{1}\right)\right) \]
6. Step-by-step derivation

7. Simplified 10.3%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{1} \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot 0 \]

   2. mul0-rgt 10.3%

      \[\leadsto 0 \]

9. Applied egg-rr 10.3%

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

Reproduce

herbie shell --seed 2024141 
(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))))