Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 18.6s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Final simplification 99.1%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} + u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{1}{-1 - e^{\frac{\pi}{s}}}\right)} + -1\right) \]
4. 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 99.1%

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

3. Simplified 99.1%

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

Alternative 3: 25.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\log s - \log \pi\right) + u \cdot \left(s \cdot 2 + u \cdot \left(s \cdot 2 + 2.6666666666666665 \cdot \left(s \cdot u\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (+
  (* s (- (log s) (log PI)))
  (* u (+ (* s 2.0) (* u (+ (* s 2.0) (* 2.6666666666666665 (* s u))))))))

C

float code(float u, float s) {
	return (s * (logf(s) - logf(((float) M_PI)))) + (u * ((s * 2.0f) + (u * ((s * 2.0f) + (2.6666666666666665f * (s * u))))));
}

Julia

function code(u, s)
	return Float32(Float32(s * Float32(log(s) - log(Float32(pi)))) + Float32(u * Float32(Float32(s * Float32(2.0)) + Float32(u * Float32(Float32(s * Float32(2.0)) + Float32(Float32(2.6666666666666665) * Float32(s * u)))))))
end

MATLAB

function tmp = code(u, s)
	tmp = (s * (log(s) - log(single(pi)))) + (u * ((s * single(2.0)) + (u * ((s * single(2.0)) + (single(2.6666666666666665) * (s * u))))));
end

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf 24.7%

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

   1. cancel-sign-sub-inv 24.7%

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

   2. metadata-eval 24.7%

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

   3. distribute-rgt-out-- 24.7%

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

   4. metadata-eval 24.7%

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

   5. *-commutative 24.7%

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

5. Simplified 24.7%

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

6. Taylor expanded in s around 0 24.6%

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

   1. mul-1-neg 24.6%

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

   2. *-commutative 24.6%

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

   3. distribute-rgt-neg-in 24.6%

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

8. Simplified 24.6%

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

9. Taylor expanded in u around 0 25.0%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \left(\log \pi - \log s\right)\right) + u \cdot \left(2 \cdot s + u \cdot \left(2 \cdot s + 2.6666666666666665 \cdot \left(s \cdot u\right)\right)\right)} \]

10. Final simplification 25.0%

    \[\leadsto s \cdot \left(\log s - \log \pi\right) + u \cdot \left(s \cdot 2 + u \cdot \left(s \cdot 2 + 2.6666666666666665 \cdot \left(s \cdot u\right)\right)\right) \]
11. Add Preprocessing

Alternative 4: 25.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf 24.7%

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

   1. cancel-sign-sub-inv 24.7%

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

   2. metadata-eval 24.7%

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

   3. distribute-rgt-out-- 24.7%

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

   4. metadata-eval 24.7%

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

   5. *-commutative 24.7%

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

5. Simplified 24.7%

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

6. Taylor expanded in s around 0 24.6%

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

   1. mul-1-neg 24.6%

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

   2. *-commutative 24.6%

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

   3. distribute-rgt-neg-in 24.6%

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

8. Simplified 24.6%

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

9. Taylor expanded in u around 0 25.0%

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

10. Final simplification 25.0%

    \[\leadsto s \cdot \left(\log s - \log \pi\right) + u \cdot \left(s \cdot 2 + 2 \cdot \left(s \cdot u\right)\right) \]
11. Add Preprocessing

Alternative 5: 25.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return fma(Float32(-s), log1p(Float32(Float32(pi) / s)), Float32(Float32(Float32(2.0) * Float32(u * Float32(pi))) / Float32(Float32(1.0) + Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf 24.7%

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

   1. cancel-sign-sub-inv 24.7%

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

   2. metadata-eval 24.7%

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

   3. distribute-rgt-out-- 24.7%

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

   4. metadata-eval 24.7%

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

   5. *-commutative 24.7%

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

5. Simplified 24.7%

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

6. Taylor expanded in u around 0 24.9%

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

   1. log1p-define 24.9%

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

   2. associate-*r* 24.9%

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

   3. fma-define 24.9%

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

   4. neg-mul-1 24.9%

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

   5. *-commutative 24.9%

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

   6. associate-*l/ 24.9%

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

   7. *-commutative 24.9%

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

   8. +-commutative 24.9%

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

8. Simplified 24.9%

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

9. Final simplification 24.9%

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

Alternative 6: 25.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	float t_0 = 1.0f + (((float) M_PI) / s);
	return (2.0f * ((u * ((float) M_PI)) / 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(2.0) * Float32(Float32(u * Float32(pi)) / t_0)) - Float32(s * log(t_0)))
end

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf 24.7%

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

   1. cancel-sign-sub-inv 24.7%

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

   2. metadata-eval 24.7%

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

   3. distribute-rgt-out-- 24.7%

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

   4. metadata-eval 24.7%

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

   5. *-commutative 24.7%

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

5. Simplified 24.7%

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

6. Taylor expanded in u around 0 24.9%

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

7. Final simplification 24.9%

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

Alternative 7: 25.1% accurate, 4.0× speedup

Math

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

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(s * Float32(-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 99.1%

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

3. Taylor expanded in s around -inf 24.7%

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

   1. cancel-sign-sub-inv 24.7%

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

   2. metadata-eval 24.7%

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

   3. distribute-rgt-out-- 24.7%

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

   4. metadata-eval 24.7%

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

   5. *-commutative 24.7%

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

5. Simplified 24.7%

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

6. Taylor expanded in u around 0 24.9%

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

   1. mul-1-neg 24.9%

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

   2. log1p-define 24.9%

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

   3. distribute-rgt-neg-in 24.9%

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

8. Simplified 24.9%

   \[\leadsto \color{blue}{s \cdot \left(-\mathsf{log1p}\left(\frac{\pi}{s}\right)\right)} \]
9. Step-by-step derivation

   1. log1p-undefine 24.9%

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

10. Applied egg-rr 24.9%

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

Alternative 8: 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 99.1%

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

3. Taylor expanded in s around -inf 24.7%

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

   1. cancel-sign-sub-inv 24.7%

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

   2. metadata-eval 24.7%

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

   3. distribute-rgt-out-- 24.7%

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

   4. metadata-eval 24.7%

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

   5. *-commutative 24.7%

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

5. Simplified 24.7%

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

6. Taylor expanded in u around 0 24.9%

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

   1. mul-1-neg 24.9%

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

   2. log1p-define 24.9%

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

   3. distribute-rgt-neg-in 24.9%

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

8. Simplified 24.9%

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

Alternative 9: 11.5% accurate, 33.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around -inf 11.8%

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

   1. associate--r+ 11.8%

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

   2. cancel-sign-sub-inv 11.8%

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

   3. metadata-eval 11.8%

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

   4. cancel-sign-sub-inv 11.8%

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

   5. associate-*r* 11.8%

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

   6. distribute-rgt-out 11.8%

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

   7. metadata-eval 11.8%

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

   8. associate-*r* 11.8%

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

7. Simplified 11.8%

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

8. Taylor expanded in u around inf 11.8%

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

9. Final simplification 11.8%

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

Alternative 10: 11.5% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf 24.7%

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

   1. cancel-sign-sub-inv 24.7%

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

   2. metadata-eval 24.7%

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

   3. distribute-rgt-out-- 24.7%

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

   4. metadata-eval 24.7%

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

   5. *-commutative 24.7%

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

5. Simplified 24.7%

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

6. Taylor expanded in s around inf 11.8%

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

   1. distribute-rgt-in 11.8%

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

   2. *-commutative 11.8%

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

   3. associate-*l* 11.8%

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

   4. metadata-eval 11.8%

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

   5. *-commutative 11.8%

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

   6. neg-mul-1 11.8%

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

   7. +-commutative 11.8%

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

   8. *-commutative 11.8%

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

   9. associate-*l* 11.8%

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

   10. *-commutative 11.8%

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

   11. metadata-eval 11.8%

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

8. Simplified 11.8%

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

9. Final simplification 11.8%

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

Alternative 11: 11.2% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf 24.7%

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

   1. cancel-sign-sub-inv 24.7%

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

   2. metadata-eval 24.7%

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

   3. distribute-rgt-out-- 24.7%

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

   4. metadata-eval 24.7%

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

   5. *-commutative 24.7%

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

5. Simplified 24.7%

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

6. Taylor expanded in u around 0 24.9%

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

   1. mul-1-neg 24.9%

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

   2. log1p-define 24.9%

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

   3. distribute-rgt-neg-in 24.9%

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

8. Simplified 24.9%

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

9. Taylor expanded in s around inf 11.6%

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

    1. div-inv 11.6%

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

11. Applied egg-rr 11.6%

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

12. Final simplification 11.6%

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

Alternative 12: 11.2% accurate, 72.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf 24.7%

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

   1. cancel-sign-sub-inv 24.7%

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

   2. metadata-eval 24.7%

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

   3. distribute-rgt-out-- 24.7%

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

   4. metadata-eval 24.7%

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

   5. *-commutative 24.7%

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

5. Simplified 24.7%

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

6. Taylor expanded in u around 0 24.9%

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

   1. mul-1-neg 24.9%

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

   2. log1p-define 24.9%

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

   3. distribute-rgt-neg-in 24.9%

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

8. Simplified 24.9%

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

9. Taylor expanded in s around inf 11.6%

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

10. Final simplification 11.6%

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

Alternative 13: 11.2% accurate, 216.5× speedup

Math

\[\begin{array}{l} \\ -\pi \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u around 0 11.6%

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

   1. mul-1-neg 11.6%

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

7. Simplified 11.6%

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

Reproduce

herbie shell --seed 2024086 
(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))))