Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 99.0%

Time: 23.4s

Alternatives: 14

Speedup: 0.4×

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: 99.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	t_0 = (u / (single(1.0) + exp((single(pi) / -s)))) + ((single(1.0) - u) / (single(1.0) + exp((single(pi) / s))));
	tmp = -s * log((((t_0 ^ single(-3.0)) + single(-1.0)) / ((t_0 ^ single(-2.0)) + (single(1.0) + (single(1.0) / t_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 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	t_0 = (u / (single(1.0) + exp((single(pi) / -s)))) + ((single(1.0) - u) / (single(1.0) + exp((single(pi) / s))));
	tmp = -s * log((single(1.0) / ((single(1.0) / ((t_0 ^ single(-2.0)) + single(-1.0))) * (single(1.0) + (single(1.0) / t_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 99.1%

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

   1. clear-num N/A

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

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

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

8. Applied egg-rr 99.0%

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

   1. flip-+ N/A

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

   2. associate-/r/ N/A

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

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

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

10. Applied egg-rr 99.1%

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

11. Final simplification 99.1%

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

Alternative 3: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

6. Applied egg-rr 99.1%

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

   1. clear-num N/A

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

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

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 4: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

6. Applied egg-rr 99.1%

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

   1. metadata-eval N/A

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

   2. pow-pow N/A

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

   3. inv-pow N/A

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

   4. metadata-eval N/A

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

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%

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

Alternative 5: 97.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf((-1.0f + ((1.0f + expf((((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(pi) / Float32(-s)))) / u))))
end

MATLAB

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

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

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

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

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

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

7. Simplified 93.5%

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

8. Taylor expanded in s around 0

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

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

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

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

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

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

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

   4. neg-sub0 N/A

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

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

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

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

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

   7. PI-lowering-PI.f32 97.8%

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

10. Simplified 97.8%

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

11. Final simplification 97.8%

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

Alternative 6: 25.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{\pi}{s}\\ u \cdot \left(2 \cdot \left(\frac{\pi}{t\_0} + \frac{\frac{u \cdot \left(\pi \cdot \pi\right)}{s}}{t\_0 \cdot t\_0}\right)\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (/ PI s))))
   (-
    (* u (* 2.0 (+ (/ PI t_0) (/ (/ (* u (* PI PI)) s) (* t_0 t_0)))))
    (* s (log1p (/ PI s))))))

C

float code(float u, float s) {
	float t_0 = 1.0f + (((float) M_PI) / s);
	return (u * (2.0f * ((((float) M_PI) / t_0) + (((u * (((float) M_PI) * ((float) M_PI))) / s) / (t_0 * t_0))))) - (s * log1pf((((float) M_PI) / s)));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) + Float32(Float32(pi) / s))
	return Float32(Float32(u * Float32(Float32(2.0) * Float32(Float32(Float32(pi) / t_0) + Float32(Float32(Float32(u * Float32(Float32(pi) * Float32(pi))) / s) / Float32(t_0 * t_0))))) - 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

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

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

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

7. Simplified 24.7%

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

8. Taylor expanded in u around 0

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

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

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

   2. associate-*r* N/A

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

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

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

   4. mul-1-neg N/A

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

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

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

   6. log1p-define N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right)\right), \left(u \cdot \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + 2 \cdot \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right)\right), \left(u \cdot \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + 2 \cdot \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI}\left(\right), s\right)\right)\right), \left(u \cdot \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + 2 \cdot \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right), \left(u \cdot \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + 2 \cdot \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right), \mathsf{*.f32}\left(u, \color{blue}{\left(2 \cdot \frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + 2 \cdot \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)}\right)\right) \]

   11. distribute-lft-out N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right), \mathsf{*.f32}\left(u, \left(2 \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)}\right)\right)\right) \]

10. Simplified 25.0%

    \[\leadsto \color{blue}{\left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) + u \cdot \left(2 \cdot \left(\frac{\pi}{1 + \frac{\pi}{s}} + \frac{\frac{u \cdot \left(\pi \cdot \pi\right)}{s}}{\left(1 + \frac{\pi}{s}\right) \cdot \left(1 + \frac{\pi}{s}\right)}\right)\right)} \]

11. Final simplification 25.0%

    \[\leadsto u \cdot \left(2 \cdot \left(\frac{\pi}{1 + \frac{\pi}{s}} + \frac{\frac{u \cdot \left(\pi \cdot \pi\right)}{s}}{\left(1 + \frac{\pi}{s}\right) \cdot \left(1 + \frac{\pi}{s}\right)}\right)\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
12. Add Preprocessing

Alternative 7: 25.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot 2\right) \cdot \left(u \cdot \left(u + 1\right)\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (- (* (* s 2.0) (* u (+ u 1.0))) (* s (log1p (/ PI s)))))

C

float code(float u, float s) {
	return ((s * 2.0f) * (u * (u + 1.0f))) - (s * log1pf((((float) M_PI) / s)));
}

Julia

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

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

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

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

7. Simplified 24.7%

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

8. Taylor expanded in u around 0

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

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

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

   2. associate-*r* N/A

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

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

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

   4. mul-1-neg N/A

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

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

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

   6. log1p-define N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right)\right), \left(u \cdot \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + 2 \cdot \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right)\right), \left(u \cdot \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + 2 \cdot \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI}\left(\right), s\right)\right)\right), \left(u \cdot \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + 2 \cdot \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right), \left(u \cdot \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + 2 \cdot \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right), \mathsf{*.f32}\left(u, \color{blue}{\left(2 \cdot \frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + 2 \cdot \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)}\right)\right) \]

   11. distribute-lft-out N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right), \mathsf{*.f32}\left(u, \left(2 \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)}\right)\right)\right) \]

10. Simplified 25.0%

    \[\leadsto \color{blue}{\left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) + u \cdot \left(2 \cdot \left(\frac{\pi}{1 + \frac{\pi}{s}} + \frac{\frac{u \cdot \left(\pi \cdot \pi\right)}{s}}{\left(1 + \frac{\pi}{s}\right) \cdot \left(1 + \frac{\pi}{s}\right)}\right)\right)} \]

11. Taylor expanded in s around 0

    \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right), \color{blue}{\left(2 \cdot \left(s \cdot \left(u \cdot \left(1 + u\right)\right)\right)\right)}\right) \]
12. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right), \left(\left(2 \cdot s\right) \cdot \color{blue}{\left(u \cdot \left(1 + u\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right), \mathsf{*.f32}\left(\left(2 \cdot s\right), \color{blue}{\left(u \cdot \left(1 + u\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(2, s\right), \left(\color{blue}{u} \cdot \left(1 + u\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(2, s\right), \mathsf{*.f32}\left(u, \color{blue}{\left(1 + u\right)}\right)\right)\right) \]

    5. +-lowering-+.f32 25.0%

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(2, s\right), \mathsf{*.f32}\left(u, \mathsf{+.f32}\left(1, \color{blue}{u}\right)\right)\right)\right) \]

13. Simplified 25.0%

    \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) + \color{blue}{\left(2 \cdot s\right) \cdot \left(u \cdot \left(1 + u\right)\right)} \]

14. Final simplification 25.0%

    \[\leadsto \left(s \cdot 2\right) \cdot \left(u \cdot \left(u + 1\right)\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
15. Add Preprocessing

Alternative 8: 25.0% 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

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

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

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

7. Simplified 24.7%

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

8. Taylor expanded in u around 0

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

   1. associate-*r* N/A

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

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

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

   3. mul-1-neg N/A

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

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

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

   5. log1p-define N/A

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

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

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

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

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

   8. PI-lowering-PI.f32 25.0%

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

10. Simplified 25.0%

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

Alternative 9: 14.2% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(0.0) - ((s * s) / 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 10.7%

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

7. Simplified 10.7%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\pi}{s}} \]
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 10.7%

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

9. Applied egg-rr 10.7%

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

    1. neg-sub0 N/A

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

    2. flip-- N/A

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

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

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

    4. metadata-eval N/A

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

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

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

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

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

    7. +-lowering-+.f32 13.8%

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

11. Applied egg-rr 13.8%

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

12. Final simplification 13.8%

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

Alternative 10: 14.0% accurate, 39.4× speedup

Math

\[\begin{array}{l} \\ \frac{\pi}{s} \cdot \left(0 - \frac{s \cdot s}{s}\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(pi) / s) * (single(0.0) - ((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 10.7%

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

7. Simplified 10.7%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\pi}{s}} \]
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. metadata-eval N/A

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

   4. neg-sub0 N/A

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

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

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

   6. distribute-rgt-neg-in N/A

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

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

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

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

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

   9. +-lowering-+.f32 13.7%

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

9. Applied egg-rr 13.7%

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

10. Final simplification 13.7%

    \[\leadsto \frac{\pi}{s} \cdot \left(0 - \frac{s \cdot s}{s}\right) \]
11. Add Preprocessing

Alternative 11: 11.7% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(pi) * single(4.0)) * ((u * single(0.5)) + 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

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

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

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

7. Simplified 24.7%

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

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

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

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

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

   3. log1p-define N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\mathsf{log1p}\left(\left(\left(\mathsf{PI}\left(\right) \cdot u\right) \cdot \frac{1}{2} + \mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right) \cdot \frac{-4}{s}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot u\right) \cdot \frac{1}{2} + \mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right) \cdot \frac{-4}{s}\right)\right)\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\left(\left(\left(\mathsf{PI}\left(\right) \cdot u\right) \cdot \frac{1}{2} + \mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right) \cdot \frac{1}{\frac{s}{-4}}\right)\right)\right) \]

   6. un-div-inv N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot u\right) \cdot \frac{1}{2} + \mathsf{PI}\left(\right) \cdot \frac{-1}{4}}{\frac{s}{-4}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{PI}\left(\right) \cdot u\right) \cdot \frac{1}{2} + \mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right), \left(\frac{s}{-4}\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{PI}\left(\right) \cdot \left(u \cdot \frac{1}{2}\right) + \mathsf{PI}\left(\right) \cdot \frac{-1}{4}\right), \left(\frac{s}{-4}\right)\right)\right)\right) \]

   9. distribute-lft-out N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{PI}\left(\right) \cdot \left(u \cdot \frac{1}{2} + \frac{-1}{4}\right)\right), \left(\frac{s}{-4}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{PI}\left(\right), \left(u \cdot \frac{1}{2} + \frac{-1}{4}\right)\right), \left(\frac{s}{-4}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \left(u \cdot \frac{1}{2} + \frac{-1}{4}\right)\right), \left(\frac{s}{-4}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\left(u \cdot \frac{1}{2}\right), \frac{-1}{4}\right)\right), \left(\frac{s}{-4}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \frac{1}{2}\right), \frac{-1}{4}\right)\right), \left(\frac{s}{-4}\right)\right)\right)\right) \]

   14. /-lowering-/.f32 24.8%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\mathsf{*.f32}\left(u, \frac{1}{2}\right), \frac{-1}{4}\right)\right), \mathsf{/.f32}\left(s, -4\right)\right)\right)\right) \]

9. Applied egg-rr 24.8%

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

10. Taylor expanded in s around inf

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

    1. associate-*r* N/A

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

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

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

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

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

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

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

    5. sub-neg N/A

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

    6. metadata-eval N/A

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

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

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

    8. *-commutative N/A

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

    9. *-lowering-*.f32 10.9%

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

12. Simplified 10.9%

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

13. Final simplification 10.9%

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

Alternative 12: 11.7% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(pi) * (u * single(2.0))) - 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 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. +-lowering-+.f32 N/A

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

7. Simplified 24.7%

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

8. Taylor expanded in u around 0

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

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

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

   2. associate-*r* N/A

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

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

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

   4. mul-1-neg N/A

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

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

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

   6. log1p-define N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \left(\mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right)\right), \left(u \cdot \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + 2 \cdot \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right)\right), \left(u \cdot \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + 2 \cdot \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI}\left(\right), s\right)\right)\right), \left(u \cdot \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + 2 \cdot \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right), \left(u \cdot \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + 2 \cdot \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right), \mathsf{*.f32}\left(u, \color{blue}{\left(2 \cdot \frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + 2 \cdot \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)}\right)\right) \]

   11. distribute-lft-out N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(s\right), \mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{PI.f32}\left(\right), s\right)\right)\right), \mathsf{*.f32}\left(u, \left(2 \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{1 + \frac{\mathsf{PI}\left(\right)}{s}} + \frac{u \cdot {\mathsf{PI}\left(\right)}^{2}}{s \cdot {\left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}^{2}}\right)}\right)\right)\right) \]

10. Simplified 25.0%

    \[\leadsto \color{blue}{\left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) + u \cdot \left(2 \cdot \left(\frac{\pi}{1 + \frac{\pi}{s}} + \frac{\frac{u \cdot \left(\pi \cdot \pi\right)}{s}}{\left(1 + \frac{\pi}{s}\right) \cdot \left(1 + \frac{\pi}{s}\right)}\right)\right)} \]

11. Taylor expanded in s around inf

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

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

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

    2. mul-1-neg N/A

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

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

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

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

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

    5. associate-*r* N/A

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

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

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

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

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

    8. PI-lowering-PI.f32 10.9%

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

13. Simplified 10.9%

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

14. Final simplification 10.9%

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

Alternative 13: 11.4% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(pi) * (s * (single(-1.0) / 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 10.7%

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

7. Simplified 10.7%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   8. neg-lowering-neg.f32 10.7%

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

9. Applied egg-rr 10.7%

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

10. Final simplification 10.7%

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

Alternative 14: 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 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 10.7%

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

7. Simplified 10.7%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (u s)
  :name "Sample trimmed logistic on [-pi, pi]"
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0)) (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- s) (log (- (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) 1.0))))