Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 20.3s

Alternatives: 14

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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ t_1 := e^{\frac{\pi}{-s}}\\ t_2 := \frac{u}{1 + t\_1} + \frac{1 - u}{1 + t\_0}\\ s \cdot \left(-\log \left(\frac{-1 + {t\_2}^{-3}}{{t\_2}^{-2} + \left(1 + \frac{-1}{\frac{u}{-1 - t\_1} + \frac{1 - u}{-1 - t\_0}}\right)}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s)))
        (t_1 (exp (/ PI (- s))))
        (t_2 (+ (/ u (+ 1.0 t_1)) (/ (- 1.0 u) (+ 1.0 t_0)))))
   (*
    s
    (-
     (log
      (/
       (+ -1.0 (pow t_2 -3.0))
       (+
        (pow t_2 -2.0)
        (+
         1.0
         (/ -1.0 (+ (/ u (- -1.0 t_1)) (/ (- 1.0 u) (- -1.0 t_0))))))))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	float t_1 = expf((((float) M_PI) / -s));
	float t_2 = (u / (1.0f + t_1)) + ((1.0f - u) / (1.0f + t_0));
	return s * -logf(((-1.0f + powf(t_2, -3.0f)) / (powf(t_2, -2.0f) + (1.0f + (-1.0f / ((u / (-1.0f - t_1)) + ((1.0f - u) / (-1.0f - t_0))))))));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	t_1 = exp(Float32(Float32(pi) / Float32(-s)))
	t_2 = Float32(Float32(u / Float32(Float32(1.0) + t_1)) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + t_0)))
	return Float32(s * Float32(-log(Float32(Float32(Float32(-1.0) + (t_2 ^ Float32(-3.0))) / Float32((t_2 ^ Float32(-2.0)) + Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(u / Float32(Float32(-1.0) - t_1)) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(-1.0) - t_0))))))))))
end

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s));
	t_1 = exp((single(pi) / -s));
	t_2 = (u / (single(1.0) + t_1)) + ((single(1.0) - u) / (single(1.0) + t_0));
	tmp = s * -log(((single(-1.0) + (t_2 ^ single(-3.0))) / ((t_2 ^ single(-2.0)) + (single(1.0) + (single(-1.0) / ((u / (single(-1.0) - t_1)) + ((single(1.0) - u) / (single(-1.0) - t_0))))))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. clear-num 98.8%

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

   2. associate-/r/ 98.8%

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

6. Applied egg-rr 98.8%

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

   1. flip3-+ 98.8%

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

8. Applied egg-rr 98.9%

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

   1. +-commutative 98.9%

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

   2. associate-*l/ 98.9%

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

   3. metadata-eval 98.9%

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

10. Simplified 98.9%

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

11. Final simplification 98.9%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 3: 25.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Taylor expanded in s around -inf 24.8%

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

6. Taylor expanded in u around 0 25.1%

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

   1. mul-1-neg 25.1%

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

   2. unsub-neg 25.1%

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

8. Simplified 25.1%

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

Alternative 4: 25.2% 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 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.8%

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

6. Taylor expanded in u around -inf 21.9%

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

   1. mul-1-neg 21.9%

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

   2. distribute-rgt-neg-in 21.9%

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

   3. +-commutative 21.9%

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

   4. mul-1-neg 21.9%

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

   5. unsub-neg 21.9%

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

   6. distribute-rgt-out-- 21.9%

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

   7. metadata-eval 21.9%

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

8. Simplified 21.9%

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

9. Taylor expanded in u around 0 25.1%

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

10. Final simplification 25.1%

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

Alternative 5: 25.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Taylor expanded in s around -inf 24.8%

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

6. Taylor expanded in u around -inf 21.9%

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

   1. mul-1-neg 21.9%

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

   2. distribute-rgt-neg-in 21.9%

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

   3. +-commutative 21.9%

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

   4. mul-1-neg 21.9%

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

   5. unsub-neg 21.9%

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

   6. distribute-rgt-out-- 21.9%

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

   7. metadata-eval 21.9%

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

8. Simplified 21.9%

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

9. Taylor expanded in u around 0 25.1%

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

    1. log1p-define 25.1%

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

    2. associate-/l* 25.1%

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

    3. associate-/r* 25.1%

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

11. Simplified 25.1%

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

12. Final simplification 25.1%

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

Alternative 6: 25.2% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.8%

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

6. Taylor expanded in u around 0 25.1%

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

   1. *-commutative 25.1%

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

8. Simplified 25.1%

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

9. Final simplification 25.1%

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

Alternative 7: 25.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.8%

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

6. Taylor expanded in u around 0 25.1%

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

   1. associate-*r* 25.1%

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

   2. neg-mul-1 25.1%

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

   3. log1p-define 25.1%

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

8. Simplified 25.1%

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

Alternative 8: 13.0% accurate, 20.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

5. Taylor expanded in s around -inf 24.8%

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

6. Taylor expanded in s around 0 24.8%

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

7. Taylor expanded in s around inf 13.0%

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

8. Final simplification 13.0%

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

Alternative 9: 11.8% accurate, 28.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

5. Taylor expanded in s around -inf 12.6%

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

   1. associate--r+ 12.6%

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

   2. cancel-sign-sub-inv 12.6%

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

   3. cancel-sign-sub-inv 12.6%

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

   4. metadata-eval 12.6%

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

   5. associate-*r* 12.6%

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

   6. distribute-rgt-out 12.6%

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

   7. metadata-eval 12.6%

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

   8. *-commutative 12.6%

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

   9. *-commutative 12.6%

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

   10. associate-*l* 12.6%

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

7. Simplified 12.6%

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

8. Final simplification 12.6%

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

Alternative 10: 11.8% 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%

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

5. Taylor expanded in s around -inf 12.6%

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

   1. associate--r+ 12.6%

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

   2. cancel-sign-sub-inv 12.6%

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

   3. cancel-sign-sub-inv 12.6%

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

   4. metadata-eval 12.6%

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

   5. associate-*r* 12.6%

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

   6. distribute-rgt-out 12.6%

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

   7. metadata-eval 12.6%

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

   8. *-commutative 12.6%

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

   9. *-commutative 12.6%

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

   10. associate-*l* 12.6%

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

7. Simplified 12.6%

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

8. Taylor expanded in u around 0 12.6%

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

   1. associate-*r* 12.6%

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

   2. distribute-rgt-out 12.6%

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

10. Simplified 12.6%

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

11. Final simplification 12.6%

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

Alternative 11: 11.8% 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 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 12.6%

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

   1. associate--r+ 12.6%

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

   2. cancel-sign-sub-inv 12.6%

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

   3. cancel-sign-sub-inv 12.6%

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

   4. metadata-eval 12.6%

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

   5. associate-*r* 12.6%

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

   6. distribute-rgt-out 12.6%

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

   7. metadata-eval 12.6%

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

   8. *-commutative 12.6%

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

   9. *-commutative 12.6%

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

   10. associate-*l* 12.6%

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

7. Simplified 12.6%

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

8. Taylor expanded in u around inf 12.6%

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

9. Taylor expanded in u around 0 12.6%

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

    1. +-commutative 12.6%

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

    2. mul-1-neg 12.6%

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

    3. unsub-neg 12.6%

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

    4. *-commutative 12.6%

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

    5. *-commutative 12.6%

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

11. Simplified 12.6%

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

12. Final simplification 12.6%

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

Alternative 12: 11.5% accurate, 72.2× speedup

Math

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

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

MATLAB

function tmp = code(u, s)
	tmp = s * (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%

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

5. Taylor expanded in u around 0 12.3%

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

6. Final simplification 12.3%

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

Alternative 13: 11.6% accurate, 216.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in u around 0 12.3%

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

   1. neg-mul-1 12.3%

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

7. Simplified 12.3%

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

Alternative 14: 10.3% accurate, 433.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 0.0)

C

float code(float u, float s) {
	return 0.0f;
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

function code(u, s)
	return Float32(0.0)
end

MATLAB

function tmp = code(u, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 10.2%

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

6. Taylor expanded in s around 0 10.2%

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

Reproduce

herbie shell --seed 2024169 
(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))))