Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.8%

Time: 23.2s

Alternatives: 9

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. add-cube-cbrt 98.8%

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

   2. fma-def 98.8%

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

5. Applied egg-rr 98.9%

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

6. Final simplification 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(\sqrt[3]{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2}}, \frac{1}{\sqrt[3]{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}, -1\right)\right) \]

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	float t_0 = 1.0f + 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 / t_0)))))));
}

Julia

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

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) + 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) / t_0)))))));
end

Derivation

1. Initial program 98.9%

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

2. Final simplification 98.9%

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

Alternative 3: 98.9% accurate, 1.4× 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 98.9%

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

3. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 4: 27.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left({\left(1 - \frac{\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + -0.25 \cdot \left(u \cdot \pi\right)\right) \cdot -12}{s}\right)}^{0.3333333333333333}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (pow
    (- 1.0 (/ (* (+ (* PI (+ (* u -0.25) 0.25)) (* -0.25 (* u PI))) -12.0) s))
    0.3333333333333333))))

C

float code(float u, float s) {
	return -s * logf(powf((1.0f - ((((((float) M_PI) * ((u * -0.25f) + 0.25f)) + (-0.25f * (u * ((float) M_PI)))) * -12.0f) / s)), 0.3333333333333333f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log((Float32(Float32(1.0) - Float32(Float32(Float32(Float32(Float32(pi) * Float32(Float32(u * Float32(-0.25)) + Float32(0.25))) + Float32(Float32(-0.25) * Float32(u * Float32(pi)))) * Float32(-12.0)) / s)) ^ Float32(0.3333333333333333))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) - ((((single(pi) * ((u * single(-0.25)) + single(0.25))) + (single(-0.25) * (u * single(pi)))) * single(-12.0)) / s)) ^ single(0.3333333333333333)));
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%

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

   1. add-cbrt-cube 98.9%

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

   2. pow1/3 98.6%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left({\left(\left(\left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \cdot \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right) \cdot \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)}^{0.3333333333333333}\right)}\right) \]

   3. pow3 98.6%

      \[\leadsto s \cdot \left(-\log \left({\color{blue}{\left({\left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)}^{3}\right)}}^{0.3333333333333333}\right)\right) \]

5. Applied egg-rr 98.6%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left({\left({\left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)}^{3}\right)}^{0.3333333333333333}\right)}\right) \]

6. Taylor expanded in s around -inf 27.6%

   \[\leadsto s \cdot \left(-\log \left({\color{blue}{\left(1 + -1 \cdot \frac{-8 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right) + -4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s}\right)}}^{0.3333333333333333}\right)\right) \]
7. Step-by-step derivation

   1. mul-1-neg 27.6%

      \[\leadsto s \cdot \left(-\log \left({\left(1 + \color{blue}{\left(-\frac{-8 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right) + -4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s}\right)}\right)}^{0.3333333333333333}\right)\right) \]

   2. unsub-neg 27.6%

      \[\leadsto s \cdot \left(-\log \left({\color{blue}{\left(1 - \frac{-8 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right) + -4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)\right)}{s}\right)}}^{0.3333333333333333}\right)\right) \]

8. Simplified 27.6%

   \[\leadsto s \cdot \left(-\log \left({\color{blue}{\left(1 - \frac{\left(\pi \cdot \left(-0.25 \cdot u + 0.25\right) + \left(\pi \cdot u\right) \cdot -0.25\right) \cdot -12}{s}\right)}}^{0.3333333333333333}\right)\right) \]

9. Final simplification 27.6%

   \[\leadsto \left(-s\right) \cdot \log \left({\left(1 - \frac{\left(\pi \cdot \left(u \cdot -0.25 + 0.25\right) + -0.25 \cdot \left(u \cdot \pi\right)\right) \cdot -12}{s}\right)}^{0.3333333333333333}\right) \]

Alternative 5: 27.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left({\left(\mathsf{fma}\left(\frac{\pi \cdot \left(-0.25 + u \cdot 0.5\right)}{s}, -12, 1\right)\right)}^{0.3333333333333333}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (pow (fma (/ (* PI (+ -0.25 (* u 0.5))) s) -12.0 1.0) 0.3333333333333333))))

C

float code(float u, float s) {
	return -s * logf(powf(fmaf(((((float) M_PI) * (-0.25f + (u * 0.5f))) / s), -12.0f, 1.0f), 0.3333333333333333f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log((fma(Float32(Float32(Float32(pi) * Float32(Float32(-0.25) + Float32(u * Float32(0.5)))) / s), Float32(-12.0), Float32(1.0)) ^ Float32(0.3333333333333333))))
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%

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

   1. add-cbrt-cube 98.9%

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

   2. pow1/3 98.6%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left({\left(\left(\left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) \cdot \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right) \cdot \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)}^{0.3333333333333333}\right)}\right) \]

   3. pow3 98.6%

      \[\leadsto s \cdot \left(-\log \left({\color{blue}{\left({\left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)}^{3}\right)}}^{0.3333333333333333}\right)\right) \]

5. Applied egg-rr 98.6%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left({\left({\left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)}^{3}\right)}^{0.3333333333333333}\right)}\right) \]

6. Taylor expanded in s around inf 27.6%

   \[\leadsto s \cdot \left(-\log \left({\color{blue}{\left(1 + \left(-8 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s} + -4 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s}\right)\right)}}^{0.3333333333333333}\right)\right) \]
7. Step-by-step derivation

   1. +-commutative 27.6%

      \[\leadsto s \cdot \left(-\log \left({\color{blue}{\left(\left(-8 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s} + -4 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s}\right) + 1\right)}}^{0.3333333333333333}\right)\right) \]

   2. distribute-rgt-out 27.6%

      \[\leadsto s \cdot \left(-\log \left({\left(\color{blue}{\frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s} \cdot \left(-8 + -4\right)} + 1\right)}^{0.3333333333333333}\right)\right) \]

   3. fma-def 27.6%

      \[\leadsto s \cdot \left(-\log \left({\color{blue}{\left(\mathsf{fma}\left(\frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s}, -8 + -4, 1\right)\right)}}^{0.3333333333333333}\right)\right) \]

8. Simplified 27.6%

   \[\leadsto s \cdot \left(-\log \left({\color{blue}{\left(\mathsf{fma}\left(\frac{\pi \cdot \left(u \cdot 0.5 + -0.25\right)}{s}, -12, 1\right)\right)}}^{0.3333333333333333}\right)\right) \]

9. Final simplification 27.6%

   \[\leadsto \left(-s\right) \cdot \log \left({\left(\mathsf{fma}\left(\frac{\pi \cdot \left(-0.25 + u \cdot 0.5\right)}{s}, -12, 1\right)\right)}^{0.3333333333333333}\right) \]

Alternative 6: 25.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\log s - \log \left(4 \cdot \left(\pi \cdot \mathsf{fma}\left(u, 0.5, 0.25\right)\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around -inf 25.0%

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

   1. cancel-sign-sub-inv 25.0%

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

   2. metadata-eval 25.0%

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

   3. distribute-rgt-out-- 25.0%

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

   4. metadata-eval 25.0%

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

   5. *-commutative 25.0%

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

4. Simplified 25.0%

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

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 25.1%

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

   3. pow2 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{\sqrt{\color{blue}{{\left(u \cdot \left(\pi \cdot -0.5\right)\right)}^{2}}} + \pi \cdot 0.25}{s}\right) \]

6. Applied egg-rr 25.1%

   \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{\color{blue}{\sqrt{{\left(u \cdot \left(\pi \cdot -0.5\right)\right)}^{2}}} + \pi \cdot 0.25}{s}\right) \]
7. Step-by-step derivation

   1. unpow2 25.1%

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

   2. rem-sqrt-square 25.1%

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

8. Simplified 25.1%

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

9. Taylor expanded in s around 0 25.2%

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

    1. mul-1-neg 25.2%

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

    2. *-commutative 25.2%

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

    3. distribute-rgt-neg-in 25.2%

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

11. Simplified 25.2%

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

12. Final simplification 25.2%

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

Alternative 7: 25.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around -inf 25.0%

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

   1. cancel-sign-sub-inv 25.0%

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

   2. metadata-eval 25.0%

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

   3. distribute-rgt-out-- 25.0%

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

   4. metadata-eval 25.0%

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

   5. *-commutative 25.0%

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

4. Simplified 25.0%

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

5. Taylor expanded in u around 0 25.1%

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

   1. *-commutative 25.1%

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

7. Simplified 25.1%

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

8. Final simplification 25.1%

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

Alternative 8: 11.5% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

4. Taylor expanded in s around inf 11.2%

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

   1. associate--r+ 11.2%

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

   2. cancel-sign-sub-inv 11.2%

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

   3. distribute-rgt-out-- 11.2%

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

   4. *-commutative 11.2%

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

   5. metadata-eval 11.2%

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

   6. metadata-eval 11.2%

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

   7. *-commutative 11.2%

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

6. Simplified 11.2%

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

7. Taylor expanded in u around 0 11.2%

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

   1. +-commutative 11.2%

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

   2. *-commutative 11.2%

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

   3. *-commutative 11.2%

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

   4. *-commutative 11.2%

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

   5. associate-*r* 11.2%

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

   6. distribute-lft-out 11.2%

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

9. Simplified 11.2%

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

10. Final simplification 11.2%

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

Alternative 9: 11.3% accurate, 7.2× 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%

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

4. Taylor expanded in u around 0 11.1%

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

   1. neg-mul-1 11.1%

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

6. Simplified 11.1%

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

7. Final simplification 11.1%

   \[\leadsto -\pi \]

Reproduce

herbie shell --seed 2023319 
(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))))