Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 98.9%

Time: 14.1s

Alternatives: 12

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(-s\right)\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (log1p (expm1 (- s)))
  (log
   (+
    (/
     1.0
     (+ (/ u (+ 1.0 (exp (/ PI (- s))))) (/ (- 1.0 u) (+ 1.0 (exp (/ PI s))))))
    -1.0))))

C

float code(float u, float s) {
	return log1pf(expm1f(-s)) * logf(((1.0f / ((u / (1.0f + expf((((float) M_PI) / -s)))) + ((1.0f - u) / (1.0f + expf((((float) M_PI) / s)))))) + -1.0f));
}

Julia

function code(u, s)
	return Float32(log1p(expm1(Float32(-s))) * 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) / s)))))) + Float32(-1.0))))
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

   1. add-sqr-sqrt 98.0%

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

   2. distribute-rgt-neg-in 98.0%

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

6. Applied egg-rr 98.0%

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

   1. distribute-rgt-neg-out 98.0%

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

   2. add-sqr-sqrt 98.9%

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

   3. log1p-expm1-u 98.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-s\right)\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]

8. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-s\right)\right)} \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]
9. Add Preprocessing

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

Alternative 3: 25.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{\pi}{s}\\ -4 \cdot \frac{s \cdot \left(u \cdot \left(\frac{\pi}{s} \cdot -0.25 - \frac{\pi}{s} \cdot 0.25\right)\right)}{t\_0} - s \cdot \log t\_0 \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (/ PI s))))
   (-
    (* -4.0 (/ (* s (* u (- (* (/ PI s) -0.25) (* (/ PI s) 0.25)))) t_0))
    (* s (log t_0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f + (((float) M_PI) / s);
	return (-4.0f * ((s * (u * (((((float) M_PI) / s) * -0.25f) - ((((float) M_PI) / s) * 0.25f)))) / t_0)) - (s * logf(t_0));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) + Float32(Float32(pi) / s))
	return Float32(Float32(Float32(-4.0) * Float32(Float32(s * Float32(u * Float32(Float32(Float32(Float32(pi) / s) * Float32(-0.25)) - Float32(Float32(Float32(pi) / s) * Float32(0.25))))) / t_0)) - Float32(s * log(t_0)))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) + (single(pi) / s);
	tmp = (single(-4.0) * ((s * (u * (((single(pi) / s) * single(-0.25)) - ((single(pi) / s) * single(0.25))))) / t_0)) - (s * log(t_0));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 24.9%

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

6. Taylor expanded in u around 0 25.0%

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

7. Final simplification 25.0%

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

Alternative 4: 25.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\frac{4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right)}{s}}{-1 - \frac{\pi}{s}} - \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (- (/ (* 4.0 (/ (* u (* PI -0.5)) s)) (- -1.0 (/ PI s))) (log1p (/ PI s)))))

C

float code(float u, float s) {
	return s * (((4.0f * ((u * (((float) M_PI) * -0.5f)) / s)) / (-1.0f - (((float) M_PI) / s))) - log1pf((((float) M_PI) / s)));
}

Julia

function code(u, s)
	return Float32(s * Float32(Float32(Float32(Float32(4.0) * Float32(Float32(u * Float32(Float32(pi) * Float32(-0.5))) / s)) / Float32(Float32(-1.0) - Float32(Float32(pi) / s))) - log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 24.9%

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

   1. expm1-log1p-u 24.9%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi\right)\right)} + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right) \]

   2. expm1-undefine 24.9%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\pi\right)} - 1\right)} + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right) \]

7. Applied egg-rr 24.9%

   \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\pi\right)} - 1\right)} + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right) \]
8. Step-by-step derivation

   1. expm1-define 24.9%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi\right)\right)} + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right) \]

9. Simplified 24.9%

   \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi\right)\right)} + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right) \]

10. Taylor expanded in u around 0 25.0%

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

    1. log1p-define 25.0%

       \[\leadsto \left(-s\right) \cdot \left(\color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} + 4 \cdot \frac{u \cdot \left(-0.25 \cdot \frac{\pi}{s} - 0.25 \cdot \frac{\pi}{s}\right)}{1 + \frac{\pi}{s}}\right) \]

    2. associate-*r/ 25.0%

       \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \color{blue}{\frac{4 \cdot \left(u \cdot \left(-0.25 \cdot \frac{\pi}{s} - 0.25 \cdot \frac{\pi}{s}\right)\right)}{1 + \frac{\pi}{s}}}\right) \]

    3. associate-*r/ 25.0%

       \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \left(\color{blue}{\frac{-0.25 \cdot \pi}{s}} - 0.25 \cdot \frac{\pi}{s}\right)\right)}{1 + \frac{\pi}{s}}\right) \]

    4. associate-*r/ 25.0%

       \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \left(\frac{-0.25 \cdot \pi}{s} - \color{blue}{\frac{0.25 \cdot \pi}{s}}\right)\right)}{1 + \frac{\pi}{s}}\right) \]

    5. div-sub 25.0%

       \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \color{blue}{\frac{-0.25 \cdot \pi - 0.25 \cdot \pi}{s}}\right)}{1 + \frac{\pi}{s}}\right) \]

    6. associate-/l* 25.0%

       \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \color{blue}{\frac{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right)}{s}}}{1 + \frac{\pi}{s}}\right) \]

    7. distribute-rgt-out-- 25.0%

       \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \frac{u \cdot \color{blue}{\left(\pi \cdot \left(-0.25 - 0.25\right)\right)}}{s}}{1 + \frac{\pi}{s}}\right) \]

    8. metadata-eval 25.0%

       \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right)}{s}}{1 + \frac{\pi}{s}}\right) \]

12. Simplified 25.0%

    \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right)}{s}}{1 + \frac{\pi}{s}}\right)} \]

13. Final simplification 25.0%

    \[\leadsto s \cdot \left(\frac{4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right)}{s}}{-1 - \frac{\pi}{s}} - \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \]
14. Add Preprocessing

Alternative 5: 25.1% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 24.9%

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

6. Taylor expanded in u around 0 25.0%

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

   1. *-commutative 25.0%

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

8. Simplified 25.0%

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

9. Final simplification 25.0%

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

Alternative 6: 25.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 24.9%

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

   1. expm1-log1p-u 24.9%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi\right)\right)} + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right) \]

   2. expm1-undefine 24.9%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\pi\right)} - 1\right)} + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right) \]

7. Applied egg-rr 24.9%

   \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\pi\right)} - 1\right)} + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right) \]
8. Step-by-step derivation

   1. expm1-define 24.9%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi\right)\right)} + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right) \]

9. Simplified 24.9%

   \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi\right)\right)} + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right) \]

10. Taylor expanded in u around 0 25.0%

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

Alternative 7: 25.1% accurate, 4.1× speedup

Math

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

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(s * Float32(-log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 24.9%

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

6. Taylor expanded in u around 0 25.0%

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

   1. associate-*r* 25.0%

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

   2. neg-mul-1 25.0%

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

   3. log1p-define 25.0%

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

8. Simplified 25.0%

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

9. Final simplification 25.0%

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

Alternative 8: 11.6% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 11.1%

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

   1. associate--r+ 11.1%

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

   2. cancel-sign-sub-inv 11.1%

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

   3. cancel-sign-sub-inv 11.1%

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

   4. metadata-eval 11.1%

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

   5. associate-*r* 11.1%

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

   6. distribute-rgt-out 11.1%

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

   7. metadata-eval 11.1%

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

   8. *-commutative 11.1%

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

   9. *-commutative 11.1%

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

   10. associate-*l* 11.1%

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

7. Simplified 11.1%

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

8. Taylor expanded in u around inf 11.1%

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

9. Final simplification 11.1%

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

Alternative 9: 11.6% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 11.1%

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

   1. associate--r+ 11.1%

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

   2. cancel-sign-sub-inv 11.1%

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

   3. cancel-sign-sub-inv 11.1%

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

   4. metadata-eval 11.1%

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

   5. associate-*r* 11.1%

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

   6. distribute-rgt-out 11.1%

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

   7. metadata-eval 11.1%

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

   8. *-commutative 11.1%

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

   9. *-commutative 11.1%

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

   10. associate-*l* 11.1%

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

7. Simplified 11.1%

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

8. Taylor expanded in u around inf 11.1%

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

   1. +-commutative 11.1%

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

   2. mul-1-neg 11.1%

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

   3. unsub-neg 11.1%

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

   4. *-commutative 11.1%

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

10. Simplified 11.1%

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

Alternative 10: 11.6% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 11.1%

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

   1. associate--r+ 11.1%

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

   2. cancel-sign-sub-inv 11.1%

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

   3. cancel-sign-sub-inv 11.1%

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

   4. metadata-eval 11.1%

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

   5. associate-*r* 11.1%

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

   6. distribute-rgt-out 11.1%

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

   7. metadata-eval 11.1%

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

   8. *-commutative 11.1%

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

   9. *-commutative 11.1%

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

   10. associate-*l* 11.1%

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

7. Simplified 11.1%

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

   1. add-cube-cbrt 11.1%

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

   2. pow3 11.1%

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

   3. *-commutative 11.1%

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

9. Applied egg-rr 11.1%

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

10. Taylor expanded in u around 0 11.1%

    \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(-0.25 \cdot \pi + \pi \cdot {\left(\sqrt[3]{-0.25}\right)}^{3}\right)\right) + -1 \cdot \pi} \]
11. Step-by-step derivation

    1. mul-1-neg 11.1%

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

    2. unsub-neg 11.1%

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

12. Simplified 11.1%

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

13. Final simplification 11.1%

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

Alternative 11: 11.6% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 11.1%

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

   1. associate--r+ 11.1%

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

   2. cancel-sign-sub-inv 11.1%

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

   3. cancel-sign-sub-inv 11.1%

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

   4. metadata-eval 11.1%

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

   5. associate-*r* 11.1%

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

   6. distribute-rgt-out 11.1%

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

   7. metadata-eval 11.1%

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

   8. *-commutative 11.1%

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

   9. *-commutative 11.1%

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

   10. associate-*l* 11.1%

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

7. Simplified 11.1%

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

8. Taylor expanded in u around 0 11.1%

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

   1. neg-mul-1 11.1%

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

   2. +-commutative 11.1%

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

   3. associate-*r* 11.1%

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

   4. neg-mul-1 11.1%

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

   5. distribute-rgt-out 11.1%

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

10. Simplified 11.1%

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

11. Final simplification 11.1%

    \[\leadsto \pi \cdot \left(-1 + u \cdot 2\right) \]
12. 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u around 0 10.9%

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

   1. neg-mul-1 10.9%

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

7. Simplified 10.9%

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

Reproduce

herbie shell --seed 2024148 
(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))))