Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 23.6s

Alternatives: 13

Speedup: 0.7×

Specification

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ t_1 := e^{0 - \frac{\pi}{s}}\\ s \cdot \log \left(\frac{1 + \frac{-1}{\frac{u}{-1 - t\_1} + \frac{1 - u}{-1 - t\_0}}}{{\left(\frac{u}{1 + t\_1} + \frac{1 - u}{1 + t\_0}\right)}^{-2} + -1}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s))) (t_1 (exp (- 0.0 (/ PI s)))))
   (*
    s
    (log
     (/
      (+ 1.0 (/ -1.0 (+ (/ u (- -1.0 t_1)) (/ (- 1.0 u) (- -1.0 t_0)))))
      (+ (pow (+ (/ u (+ 1.0 t_1)) (/ (- 1.0 u) (+ 1.0 t_0))) -2.0) -1.0))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	float t_1 = expf((0.0f - (((float) M_PI) / s)));
	return s * logf(((1.0f + (-1.0f / ((u / (-1.0f - t_1)) + ((1.0f - u) / (-1.0f - t_0))))) / (powf(((u / (1.0f + t_1)) + ((1.0f - u) / (1.0f + t_0))), -2.0f) + -1.0f)));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	t_1 = exp(Float32(Float32(0.0) - Float32(Float32(pi) / s)))
	return Float32(s * log(Float32(Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(u / Float32(Float32(-1.0) - t_1)) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(-1.0) - t_0))))) / Float32((Float32(Float32(u / Float32(Float32(1.0) + t_1)) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + t_0))) ^ Float32(-2.0)) + Float32(-1.0)))))
end

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s));
	t_1 = exp((single(0.0) - (single(pi) / s)));
	tmp = s * log(((single(1.0) + (single(-1.0) / ((u / (single(-1.0) - t_1)) + ((single(1.0) - u) / (single(-1.0) - t_0))))) / ((((u / (single(1.0) + t_1)) + ((single(1.0) - u) / (single(1.0) + t_0))) ^ single(-2.0)) + 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. flip-+ N/A

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

   2. clear-num N/A

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

   3. log-rec N/A

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ t_1 := e^{0 - \frac{\pi}{s}}\\ \left(-s\right) \cdot \log \left(\frac{1 - {\left(\frac{u}{1 + t\_1} + \frac{1 - u}{1 + t\_0}\right)}^{-2}}{-1 + \frac{1}{\frac{u}{-1 - t\_1} + \frac{1 - u}{-1 - t\_0}}}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s))) (t_1 (exp (- 0.0 (/ PI s)))))
   (*
    (- s)
    (log
     (/
      (- 1.0 (pow (+ (/ u (+ 1.0 t_1)) (/ (- 1.0 u) (+ 1.0 t_0))) -2.0))
      (+ -1.0 (/ 1.0 (+ (/ u (- -1.0 t_1)) (/ (- 1.0 u) (- -1.0 t_0))))))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	float t_1 = expf((0.0f - (((float) M_PI) / s)));
	return -s * logf(((1.0f - powf(((u / (1.0f + t_1)) + ((1.0f - u) / (1.0f + t_0))), -2.0f)) / (-1.0f + (1.0f / ((u / (-1.0f - t_1)) + ((1.0f - u) / (-1.0f - t_0)))))));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	t_1 = exp(Float32(Float32(0.0) - Float32(Float32(pi) / s)))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) - (Float32(Float32(u / Float32(Float32(1.0) + t_1)) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + t_0))) ^ Float32(-2.0))) / Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(Float32(u / Float32(Float32(-1.0) - t_1)) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(-1.0) - t_0))))))))
end

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s));
	t_1 = exp((single(0.0) - (single(pi) / s)));
	tmp = -s * log(((single(1.0) - (((u / (single(1.0) + t_1)) + ((single(1.0) - u) / (single(1.0) + t_0))) ^ single(-2.0))) / (single(-1.0) + (single(1.0) / ((u / (single(-1.0) - t_1)) + ((single(1.0) - u) / (single(-1.0) - t_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. +-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. Final simplification 98.9%

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

Alternative 3: 98.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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(0.0) - Float32(Float32(pi) / s))))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(Float32(Float32(pi) / s) / Float32(Float32(1.0) / s)) * Float32(Float32(Float32(pi) / s) / Float32(pi)))))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(-1.0) + (single(1.0) / ((u / (single(1.0) + exp((single(0.0) - (single(pi) / s))))) + ((single(1.0) - u) / (single(1.0) + exp((((single(pi) / s) / (single(1.0) / s)) * ((single(pi) / s) / 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. Step-by-step derivation

   1. +-lft-identity N/A

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

   2. flip-+ N/A

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

   3. metadata-eval N/A

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

   4. sub0-neg N/A

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

   5. +-lft-identity N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-out N/A

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

   8. +-lft-identity N/A

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

   9. metadata-eval N/A

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

   10. sub0-neg N/A

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

   11. frac-2neg N/A

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 4: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf(((-1.0f / ((u / (-1.0f - expf((0.0f - (((float) M_PI) / s))))) + ((1.0f - u) / (-1.0f - expf((((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(0.0) - Float32(Float32(pi) / s))))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(-1.0) - exp(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(0.0) - (single(pi) / s))))) + ((single(1.0) - u) / (single(-1.0) - exp((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. sub0-neg N/A

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

   2. distribute-neg-frac N/A

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

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

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

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

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

   5. PI-lowering-PI.f32 98.8%

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 5: 24.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf((1.0f + ((((u * ((float) M_PI)) * 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(u * Float32(pi)) * Float32(0.5)) + Float32(Float32(pi) * Float32(-0.25))) * Float32(Float32(-4.0) / s)))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(1.0) + ((((u * single(pi)) * 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. Taylor expanded in s around inf

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

   1. metadata-eval N/A

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

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

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

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

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

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

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

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

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

7. Simplified 24.5%

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

8. Final simplification 24.5%

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

Alternative 6: 14.2% accurate, 28.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-1 \cdot \frac{-4 \cdot \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{2} \cdot \frac{-16 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s}}{s}\right)}\right) \]

7. Simplified 11.9%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{-4 \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) + \frac{-0.5 \cdot \left(-16 \cdot \left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) + \left(0 + \left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) \cdot 16\right)\right)}{s}}{-s}} \]

9. Applied egg-rr 12.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{-s}{\left(-s\right) \cdot \left(\left(\pi \cdot \left(\left(u \cdot -0.25 + 0.25\right) \cdot -4\right) + \left(u \cdot \pi\right) \cdot 1\right) + \left(\left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) \cdot 0\right) \cdot \frac{-0.5}{s}\right)}}} \]

11. Applied egg-rr 9.0%

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

    1. sqr-neg N/A

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

    2. associate-/r* N/A

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

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

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

13. Applied egg-rr 14.3%

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

Alternative 7: 14.1% accurate, 28.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-1 \cdot \frac{-4 \cdot \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{2} \cdot \frac{-16 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s}}{s}\right)}\right) \]

7. Simplified 11.9%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{-4 \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) + \frac{-0.5 \cdot \left(-16 \cdot \left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) + \left(0 + \left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) \cdot 16\right)\right)}{s}}{-s}} \]

9. Applied egg-rr 12.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{-s}{\left(-s\right) \cdot \left(\left(\pi \cdot \left(\left(u \cdot -0.25 + 0.25\right) \cdot -4\right) + \left(u \cdot \pi\right) \cdot 1\right) + \left(\left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) \cdot 0\right) \cdot \frac{-0.5}{s}\right)}}} \]

11. Applied egg-rr 9.0%

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

    1. associate-*l* N/A

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

    2. sqr-neg N/A

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

    3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

    9. +-commutative N/A

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

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

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

    11. +-commutative N/A

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

    12. *-commutative N/A

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

    13. associate-*r* N/A

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

    14. metadata-eval N/A

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

    15. *-lft-identity N/A

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

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

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

    17. *-lowering-*.f32 14.2%

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

13. Applied egg-rr 14.2%

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

Alternative 8: 11.6% accurate, 33.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-1 \cdot \frac{-4 \cdot \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{2} \cdot \frac{-16 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s}}{s}\right)}\right) \]

7. Simplified 11.9%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{-4 \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) + \frac{-0.5 \cdot \left(-16 \cdot \left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) + \left(0 + \left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) \cdot 16\right)\right)}{s}}{-s}} \]

9. Applied egg-rr 12.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{-s}{\left(-s\right) \cdot \left(\left(\pi \cdot \left(\left(u \cdot -0.25 + 0.25\right) \cdot -4\right) + \left(u \cdot \pi\right) \cdot 1\right) + \left(\left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) \cdot 0\right) \cdot \frac{-0.5}{s}\right)}}} \]

11. Applied egg-rr 9.0%

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

    1. associate-/r* N/A

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

    2. div-inv N/A

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

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

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

13. Applied egg-rr 12.0%

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

14. Final simplification 12.0%

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

Alternative 9: 11.6% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(1.0) / (single(1.0) / (single(pi) * (u + (u + 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. Taylor expanded in s around -inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-1 \cdot \frac{-4 \cdot \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{2} \cdot \frac{-16 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s}}{s}\right)}\right) \]

7. Simplified 11.9%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{-4 \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) + \frac{-0.5 \cdot \left(-16 \cdot \left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) + \left(0 + \left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) \cdot 16\right)\right)}{s}}{-s}} \]

9. Applied egg-rr 12.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{-s}{\left(-s\right) \cdot \left(\left(\pi \cdot \left(\left(u \cdot -0.25 + 0.25\right) \cdot -4\right) + \left(u \cdot \pi\right) \cdot 1\right) + \left(\left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) \cdot 0\right) \cdot \frac{-0.5}{s}\right)}}} \]

11. Applied egg-rr 9.0%

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

    1. sqr-neg N/A

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

    2. clear-num N/A

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

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

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

    4. *-commutative N/A

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

    5. associate-/r* N/A

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

    6. *-inverses N/A

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

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

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

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

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

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

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

    10. +-commutative N/A

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

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

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

    12. +-commutative N/A

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

    13. *-commutative N/A

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

    14. associate-*r* N/A

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

    15. metadata-eval N/A

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

    16. *-lft-identity N/A

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

    17. +-lowering-+.f32 12.0%

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

13. Applied egg-rr 12.0%

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

Alternative 10: 11.6% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(pi) * (u + (u + 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. Taylor expanded in s around -inf

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-1 \cdot \frac{-4 \cdot \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{2} \cdot \frac{-16 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s}}{s}\right)}\right) \]

7. Simplified 11.9%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{-4 \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) + \frac{-0.5 \cdot \left(-16 \cdot \left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) + \left(0 + \left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) \cdot 16\right)\right)}{s}}{-s}} \]

9. Applied egg-rr 12.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{-s}{\left(-s\right) \cdot \left(\left(\pi \cdot \left(\left(u \cdot -0.25 + 0.25\right) \cdot -4\right) + \left(u \cdot \pi\right) \cdot 1\right) + \left(\left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) \cdot 0\right) \cdot \frac{-0.5}{s}\right)}}} \]

11. Applied egg-rr 9.0%

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

    1. sqr-neg N/A

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

    2. associate-/l* N/A

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

    3. *-inverses N/A

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

    4. *-commutative N/A

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

    5. associate-*l* N/A

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

    6. *-un-lft-identity N/A

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

    7. *-commutative N/A

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

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

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

    9. +-commutative N/A

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

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

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

    11. +-commutative N/A

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

    12. *-commutative N/A

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

    13. associate-*r* N/A

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

    14. metadata-eval N/A

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

    15. *-lft-identity N/A

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

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

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

    17. PI-lowering-PI.f32 12.0%

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

13. Applied egg-rr 12.0%

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

14. Final simplification 12.0%

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

Alternative 11: 11.6% 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

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-1 \cdot \frac{-4 \cdot \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{2} \cdot \frac{-16 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s}}{s}\right)}\right) \]

7. Simplified 11.9%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{-4 \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) + \frac{-0.5 \cdot \left(-16 \cdot \left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) + \left(0 + \left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) \cdot 16\right)\right)}{s}}{-s}} \]

9. Applied egg-rr 12.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{-s}{\left(-s\right) \cdot \left(\left(\pi \cdot \left(\left(u \cdot -0.25 + 0.25\right) \cdot -4\right) + \left(u \cdot \pi\right) \cdot 1\right) + \left(\left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) \cdot 0\right) \cdot \frac{-0.5}{s}\right)}}} \]

11. Applied egg-rr 9.0%

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

12. Taylor expanded in s around 0

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

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

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

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

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

    3. sub-neg N/A

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

    4. metadata-eval N/A

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

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

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

    6. *-commutative N/A

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

    7. *-lowering-*.f32 12.0%

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

14. Simplified 12.0%

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

15. Final simplification 12.0%

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

Alternative 12: 11.4% 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 11.6%

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

7. Simplified 11.6%

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

Alternative 13: 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 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

   \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \color{blue}{\left(-1 \cdot \frac{-4 \cdot \left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{2} \cdot \frac{-16 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s}}{s}\right)}\right) \]

7. Simplified 11.9%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{-4 \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) + \frac{-0.5 \cdot \left(-16 \cdot \left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) + \left(0 + \left(\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right) \cdot \left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)\right)\right) \cdot 16\right)\right)}{s}}{-s}} \]

8. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{-16 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} + \frac{-1}{4} \cdot u\right)\right)}^{2} + 16 \cdot {\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} + \frac{-1}{4} \cdot u\right)\right)}^{2}}{s}} \]
9. Step-by-step derivation

   1. distribute-rgt-out N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{{\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} + \frac{-1}{4} \cdot u\right)\right)}^{2} \cdot \left(-16 + 16\right)}{s} \]

   2. metadata-eval N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{{\left(\frac{-1}{4} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} + \frac{-1}{4} \cdot u\right)\right)}^{2} \cdot 0}{s} \]

   3. mul0-rgt N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{0}{s} \]

   4. mul0-lft N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{0 \cdot {\mathsf{PI}\left(\right)}^{2}}{s} \]

   5. metadata-eval N/A

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

   6. distribute-lft1-in N/A

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

   7. associate-*r/ N/A

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

   8. distribute-lft1-in N/A

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

   9. metadata-eval N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot \left(0 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{s} \]

   10. mul0-lft N/A

       \[\leadsto \frac{\frac{-1}{2} \cdot 0}{s} \]

   11. metadata-eval N/A

       \[\leadsto \frac{0}{s} \]

   12. /-lowering-/.f32 10.2%

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

10. Simplified 10.2%

    \[\leadsto \color{blue}{\frac{0}{s}} \]
11. Step-by-step derivation

    1. div0 10.2%

       \[\leadsto 0 \]

12. Applied egg-rr 10.2%

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

Reproduce

herbie shell --seed 2024170 
(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))))