Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 13.5s

Alternatives: 12

Speedup: 1.3×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. clear-num 98.8%

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

   2. associate-/r/ 98.8%

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

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

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(s * Float32(-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 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%

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

5. Final simplification 98.8%

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

Alternative 3: 37.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. clear-num 98.8%

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

   2. associate-/r/ 98.8%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in s around inf 37.6%

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

8. Final simplification 37.6%

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

Alternative 4: 37.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

5. Taylor expanded in s around inf 37.6%

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

6. Final simplification 37.6%

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

Alternative 5: 11.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(\left(u \cdot \pi\right) \cdot -0.25 + \pi \cdot 0.25\right)\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (pow
  (cbrt (* 4.0 (- (* 0.25 (* u PI)) (+ (* (* u PI) -0.25) (* PI 0.25)))))
  3.0))

C

float code(float u, float s) {
	return powf(cbrtf((4.0f * ((0.25f * (u * ((float) M_PI))) - (((u * ((float) M_PI)) * -0.25f) + (((float) M_PI) * 0.25f))))), 3.0f);
}

Julia

function code(u, s)
	return cbrt(Float32(Float32(4.0) * Float32(Float32(Float32(0.25) * Float32(u * Float32(pi))) - Float32(Float32(Float32(u * Float32(pi)) * Float32(-0.25)) + Float32(Float32(pi) * Float32(0.25)))))) ^ Float32(3.0)
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%

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

   1. clear-num 98.8%

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

   2. associate-/r/ 98.8%

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

6. Applied egg-rr 98.8%

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

   1. add-cube-cbrt 97.2%

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

   2. pow3 97.4%

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

8. Applied egg-rr 97.4%

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

9. Taylor expanded in s around inf 11.5%

   \[\leadsto {\left(\sqrt[3]{\color{blue}{4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)\right)}}\right)}^{3} \]

10. Final simplification 11.5%

    \[\leadsto {\left(\sqrt[3]{4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(\left(u \cdot \pi\right) \cdot -0.25 + \pi \cdot 0.25\right)\right)}\right)}^{3} \]
11. Add Preprocessing

Alternative 6: 11.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) + u \cdot \left(0.25 \cdot \frac{\frac{-1}{{\left(\frac{u}{\pi}\right)}^{2}} + \pi \cdot \pi}{\pi + \frac{\pi}{u}}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  4.0
  (+
   (* 0.25 (* u PI))
   (*
    u
    (* 0.25 (/ (+ (/ -1.0 (pow (/ u PI) 2.0)) (* PI PI)) (+ PI (/ PI u))))))))

C

float code(float u, float s) {
	return 4.0f * ((0.25f * (u * ((float) M_PI))) + (u * (0.25f * (((-1.0f / powf((u / ((float) M_PI)), 2.0f)) + (((float) M_PI) * ((float) M_PI))) / (((float) M_PI) + (((float) M_PI) / u))))));
}

Julia

function code(u, s)
	return Float32(Float32(4.0) * Float32(Float32(Float32(0.25) * Float32(u * Float32(pi))) + Float32(u * Float32(Float32(0.25) * Float32(Float32(Float32(Float32(-1.0) / (Float32(u / Float32(pi)) ^ Float32(2.0))) + Float32(Float32(pi) * Float32(pi))) / Float32(Float32(pi) + Float32(Float32(pi) / u)))))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * ((single(0.25) * (u * single(pi))) + (u * (single(0.25) * (((single(-1.0) / ((u / single(pi)) ^ single(2.0))) + (single(pi) * single(pi))) / (single(pi) + (single(pi) / u))))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 11.5%

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

6. Taylor expanded in u around inf 11.5%

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

   1. +-commutative 11.5%

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

   2. metadata-eval 11.5%

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

   3. cancel-sign-sub-inv 11.5%

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

   4. distribute-lft-out-- 11.5%

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

8. Simplified 11.5%

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

   1. flip-- 11.5%

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

   2. div-sub 11.5%

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

   3. pow2 11.5%

      \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \left(\frac{\color{blue}{{\left(\frac{\pi}{u}\right)}^{2}}}{\frac{\pi}{u} + \pi} - \frac{\pi \cdot \pi}{\frac{\pi}{u} + \pi}\right)\right)\right) \]

10. Applied egg-rr 11.5%

    \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{\left(\frac{\pi}{u}\right)}^{2}}{\frac{\pi}{u} + \pi} - \frac{\pi \cdot \pi}{\frac{\pi}{u} + \pi}\right)}\right)\right) \]
11. Step-by-step derivation

    1. unpow2 11.5%

       \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \left(\frac{{\left(\frac{\pi}{u}\right)}^{2}}{\frac{\pi}{u} + \pi} - \frac{\color{blue}{{\pi}^{2}}}{\frac{\pi}{u} + \pi}\right)\right)\right) \]

    2. div-sub 11.5%

       \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \color{blue}{\frac{{\left(\frac{\pi}{u}\right)}^{2} - {\pi}^{2}}{\frac{\pi}{u} + \pi}}\right)\right) \]

12. Simplified 11.5%

    \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \color{blue}{\frac{{\left(\frac{u}{\pi}\right)}^{-2} - \pi \cdot \pi}{\pi + \frac{\pi}{u}}}\right)\right) \]
13. Step-by-step derivation

    1. metadata-eval 11.5%

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

    2. pow-prod-up 11.5%

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

    3. pow-prod-down 11.5%

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

    4. pow2 11.5%

       \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \frac{{\color{blue}{\left({\left(\frac{u}{\pi}\right)}^{2}\right)}}^{-1} - \pi \cdot \pi}{\pi + \frac{\pi}{u}}\right)\right) \]

14. Applied egg-rr 11.5%

    \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \frac{\color{blue}{{\left({\left(\frac{u}{\pi}\right)}^{2}\right)}^{-1}} - \pi \cdot \pi}{\pi + \frac{\pi}{u}}\right)\right) \]
15. Step-by-step derivation

    1. unpow-1 11.5%

       \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \frac{\color{blue}{\frac{1}{{\left(\frac{u}{\pi}\right)}^{2}}} - \pi \cdot \pi}{\pi + \frac{\pi}{u}}\right)\right) \]

16. Simplified 11.5%

    \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \frac{\color{blue}{\frac{1}{{\left(\frac{u}{\pi}\right)}^{2}}} - \pi \cdot \pi}{\pi + \frac{\pi}{u}}\right)\right) \]

17. Final simplification 11.5%

    \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) + u \cdot \left(0.25 \cdot \frac{\frac{-1}{{\left(\frac{u}{\pi}\right)}^{2}} + \pi \cdot \pi}{\pi + \frac{\pi}{u}}\right)\right) \]
18. Add Preprocessing

Alternative 7: 11.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \frac{{\left(\frac{u}{\pi}\right)}^{-2} - \pi \cdot \pi}{\pi + \frac{\pi}{u}}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  4.0
  (-
   (* 0.25 (* u PI))
   (* u (* 0.25 (/ (- (pow (/ u PI) -2.0) (* PI PI)) (+ PI (/ PI u))))))))

C

float code(float u, float s) {
	return 4.0f * ((0.25f * (u * ((float) M_PI))) - (u * (0.25f * ((powf((u / ((float) M_PI)), -2.0f) - (((float) M_PI) * ((float) M_PI))) / (((float) M_PI) + (((float) M_PI) / u))))));
}

Julia

function code(u, s)
	return Float32(Float32(4.0) * Float32(Float32(Float32(0.25) * Float32(u * Float32(pi))) - Float32(u * Float32(Float32(0.25) * Float32(Float32((Float32(u / Float32(pi)) ^ Float32(-2.0)) - Float32(Float32(pi) * Float32(pi))) / Float32(Float32(pi) + Float32(Float32(pi) / u)))))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * ((single(0.25) * (u * single(pi))) - (u * (single(0.25) * ((((u / single(pi)) ^ single(-2.0)) - (single(pi) * single(pi))) / (single(pi) + (single(pi) / u))))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 11.5%

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

6. Taylor expanded in u around inf 11.5%

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

   1. +-commutative 11.5%

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

   2. metadata-eval 11.5%

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

   3. cancel-sign-sub-inv 11.5%

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

   4. distribute-lft-out-- 11.5%

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

8. Simplified 11.5%

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

   1. flip-- 11.5%

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

   2. div-sub 11.5%

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

   3. pow2 11.5%

      \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \left(\frac{\color{blue}{{\left(\frac{\pi}{u}\right)}^{2}}}{\frac{\pi}{u} + \pi} - \frac{\pi \cdot \pi}{\frac{\pi}{u} + \pi}\right)\right)\right) \]

10. Applied egg-rr 11.5%

    \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \color{blue}{\left(\frac{{\left(\frac{\pi}{u}\right)}^{2}}{\frac{\pi}{u} + \pi} - \frac{\pi \cdot \pi}{\frac{\pi}{u} + \pi}\right)}\right)\right) \]
11. Step-by-step derivation

    1. unpow2 11.5%

       \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \left(\frac{{\left(\frac{\pi}{u}\right)}^{2}}{\frac{\pi}{u} + \pi} - \frac{\color{blue}{{\pi}^{2}}}{\frac{\pi}{u} + \pi}\right)\right)\right) \]

    2. div-sub 11.5%

       \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \color{blue}{\frac{{\left(\frac{\pi}{u}\right)}^{2} - {\pi}^{2}}{\frac{\pi}{u} + \pi}}\right)\right) \]

12. Simplified 11.5%

    \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \color{blue}{\frac{{\left(\frac{u}{\pi}\right)}^{-2} - \pi \cdot \pi}{\pi + \frac{\pi}{u}}}\right)\right) \]
13. Add Preprocessing

Alternative 8: 11.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \mathsf{fma}\left(\pi, \frac{1}{u}, -\pi\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* 4.0 (- (* 0.25 (* u PI)) (* u (* 0.25 (fma PI (/ 1.0 u) (- PI)))))))

C

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

Julia

function code(u, s)
	return Float32(Float32(4.0) * Float32(Float32(Float32(0.25) * Float32(u * Float32(pi))) - Float32(u * Float32(Float32(0.25) * fma(Float32(pi), Float32(Float32(1.0) / u), Float32(-Float32(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%

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

5. Taylor expanded in s around inf 11.5%

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

6. Taylor expanded in u around inf 11.5%

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

   1. +-commutative 11.5%

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

   2. metadata-eval 11.5%

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

   3. cancel-sign-sub-inv 11.5%

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

   4. distribute-lft-out-- 11.5%

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

8. Simplified 11.5%

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

   1. div-inv 11.5%

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

   2. fma-neg 11.5%

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

10. Applied egg-rr 11.5%

    \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - u \cdot \left(0.25 \cdot \color{blue}{\mathsf{fma}\left(\pi, \frac{1}{u}, -\pi\right)}\right)\right) \]
11. Add Preprocessing

Alternative 9: 11.6% accurate, 28.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 11.5%

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

6. Taylor expanded in u around inf 11.5%

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

   1. +-commutative 11.5%

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

   2. metadata-eval 11.5%

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

   3. cancel-sign-sub-inv 11.5%

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

   4. distribute-lft-out-- 11.5%

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

8. Simplified 11.5%

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

9. Taylor expanded in u around 0 11.5%

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

    1. +-commutative 11.5%

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

    2. associate-*r* 11.5%

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

    3. distribute-rgt-out 11.5%

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

11. Simplified 11.5%

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

12. Final simplification 11.5%

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

Alternative 10: 11.6% accurate, 33.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

5. Taylor expanded in s around inf 11.5%

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

   1. associate--r+ 11.5%

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

   2. distribute-rgt-out-- 11.5%

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

   3. metadata-eval 11.5%

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

   4. fma-neg 11.5%

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

   5. *-commutative 11.5%

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

   6. remove-double-neg 11.5%

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

   7. neg-mul-1 11.5%

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

   8. associate-*r* 11.5%

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

   9. metadata-eval 11.5%

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

   10. distribute-rgt-neg-out 11.5%

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

   11. remove-double-neg 11.5%

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

   12. *-commutative 11.5%

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

7. Simplified 11.5%

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

8. Taylor expanded in u around inf 11.5%

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

9. Final simplification 11.5%

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

Alternative 11: 11.6% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. clear-num 98.8%

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

   2. associate-/r/ 98.8%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in s around inf 11.5%

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

   1. associate--r+ 11.5%

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

   2. *-commutative 11.5%

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

   3. *-commutative 11.5%

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

   4. associate-*r* 11.5%

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

   5. *-commutative 11.5%

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

   6. *-commutative 11.5%

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

   7. cancel-sign-sub-inv 11.5%

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

   8. associate-*r* 11.5%

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

   9. distribute-lft-out-- 11.5%

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

   10. metadata-eval 11.5%

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

   11. metadata-eval 11.5%

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

   12. *-commutative 11.5%

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

   13. *-commutative 11.5%

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

   14. +-commutative 11.5%

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

9. Simplified 11.5%

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

10. Final simplification 11.5%

    \[\leadsto 4 \cdot \left(\pi \cdot \left(-0.25 + u \cdot 0.5\right)\right) \]
11. 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%

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

5. Taylor expanded in u around 0 11.3%

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

   1. neg-mul-1 11.3%

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

7. Simplified 11.3%

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

Reproduce

herbie shell --seed 2024096 
(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))))