Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 19.9s

Alternatives: 11

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, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

5. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-+.f32 N/A

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

   4. associate-+l+ N/A

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

   5. lower-+.f32 N/A

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

8. Applied rewrites 99.0%

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

9. Final simplification 99.0%

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

Alternative 3: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \frac{1}{\left(\frac{u}{-2 + \frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{s}, \left(\pi \cdot \pi\right) \cdot -0.5\right)}{s} - \pi}{s}} + \frac{u}{1 + e^{\frac{\pi}{-s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (/
     1.0
     (+
      (+
       (/
        u
        (+
         -2.0
         (/
          (-
           (/
            (fma
             -0.16666666666666666
             (/ (* PI (* PI PI)) s)
             (* (* PI PI) -0.5))
            s)
           PI)
          s)))
       (/ u (+ 1.0 (exp (/ PI (- s))))))
      (/ 1.0 (+ 1.0 (exp (/ PI s))))))))))

C

float code(float u, float s) {
	return -s * logf((-1.0f + (1.0f / (((u / (-2.0f + (((fmaf(-0.16666666666666666f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) / s), ((((float) M_PI) * ((float) M_PI)) * -0.5f)) / s) - ((float) M_PI)) / s))) + (u / (1.0f + expf((((float) M_PI) / -s))))) + (1.0f / (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(Float32(u / Float32(Float32(-2.0) + Float32(Float32(Float32(fma(Float32(-0.16666666666666666), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) / s), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-0.5))) / s) - Float32(pi)) / s))) + Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s)))))) + Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))))))
end

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in s around -inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \left(\frac{u}{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} - \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s} - -1 \cdot \mathsf{PI}\left(\right)}{s} - 2}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}\right)} - 1\right) \]
8. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \left(\frac{u}{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} - \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s} - -1 \cdot \mathsf{PI}\left(\right)}{s} + \left(\mathsf{neg}\left(2\right)\right)}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}\right)} - 1\right) \]

   2. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \left(\frac{u}{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} - \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s} - -1 \cdot \mathsf{PI}\left(\right)}{s} + \color{blue}{-2}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}\right)} - 1\right) \]

   3. lower-+.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}} + \left(\frac{u}{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} - \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s} - -1 \cdot \mathsf{PI}\left(\right)}{s} + -2}} + \frac{u}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}}\right)} - 1\right) \]

9. Applied rewrites 98.2%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} + \left(\frac{u}{\color{blue}{\frac{-\left(-\left(\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{s}, -0.5 \cdot \left(\pi \cdot \pi\right)\right)}{s} - \pi\right)\right)}{s} + -2}} + \frac{u}{1 + e^{\frac{\pi}{-s}}}\right)} - 1\right) \]

10. Final simplification 98.2%

    \[\leadsto \left(-s\right) \cdot \log \left(-1 + \frac{1}{\left(\frac{u}{-2 + \frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{s}, \left(\pi \cdot \pi\right) \cdot -0.5\right)}{s} - \pi}{s}} + \frac{u}{1 + e^{\frac{\pi}{-s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}}\right) \]
11. Add Preprocessing

Alternative 4: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf((-1.0f + (1.0f / (u * ((1.0f / (1.0f + expf((((float) M_PI) / -s)))) + (1.0f / (-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(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(1.0) / 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) / (single(1.0) + exp((single(pi) / -s)))) + (single(1.0) / (single(-1.0) - exp((single(pi) / s)))))))));
end

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u around inf

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-exp.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. mul-1-neg N/A

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

   10. lower-/.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f32 N/A

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

   14. distribute-neg-frac N/A

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

5. Applied rewrites 97.9%

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

6. Final simplification 97.9%

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

Alternative 5: 25.1% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around -inf

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

   1. associate-*r/ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-fma.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right), \frac{4}{s}, 1\right)\right)} \]

5. Applied rewrites 25.1%

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

6. Taylor expanded in u around 0

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\mathsf{fma}\left(\frac{1}{4} \cdot \mathsf{PI}\left(\right), \frac{\color{blue}{4}}{s}, 1\right)\right) \]
7. Step-by-step derivation

8. Applied rewrites 25.3%

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

Alternative 6: 25.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around -inf

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

   1. associate-*r/ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-fma.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right), \frac{4}{s}, 1\right)\right)} \]

5. Applied rewrites 25.1%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 25.3%

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

Alternative 7: 11.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(0.25 - u \cdot -0.5\right)\\ \frac{\left(\pi \cdot \mathsf{fma}\left(u, -0.5, 0.25\right)\right) \cdot t\_0}{t\_0} \cdot -4 \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* PI (- 0.25 (* u -0.5)))))
   (* (/ (* (* PI (fma u -0.5 0.25)) t_0) t_0) -4.0)))

C

float code(float u, float s) {
	float t_0 = ((float) M_PI) * (0.25f - (u * -0.5f));
	return (((((float) M_PI) * fmaf(u, -0.5f, 0.25f)) * t_0) / t_0) * -4.0f;
}

Julia

function code(u, s)
	t_0 = Float32(Float32(pi) * Float32(Float32(0.25) - Float32(u * Float32(-0.5))))
	return Float32(Float32(Float32(Float32(Float32(pi) * fma(u, Float32(-0.5), Float32(0.25))) * t_0) / t_0) * Float32(-4.0))
end

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot -4 \]

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-out-- N/A

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

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{4}, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right) \cdot u\right) \cdot -4 \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{4}, \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} \cdot u\right)}\right) \cdot -4 \]

   13. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{4}, \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} \cdot u\right)}\right) \cdot -4 \]

   14. lower-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{4}, \color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{-1}{2} \cdot u\right)\right) \cdot -4 \]

   15. lower-*.f32 11.7

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

5. Applied rewrites 11.7%

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

7. Applied rewrites 11.7%

   \[\leadsto \frac{\left(\pi \cdot \mathsf{fma}\left(u, -0.5, 0.25\right)\right) \cdot \left(\pi \cdot \left(0.25 - u \cdot -0.5\right)\right)}{\pi \cdot \left(0.25 - u \cdot -0.5\right)} \cdot -4 \]
8. Add Preprocessing

Alternative 8: 11.6% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot -4 \]

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-out-- N/A

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

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{4}, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right) \cdot u\right) \cdot -4 \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{4}, \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} \cdot u\right)}\right) \cdot -4 \]

   13. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{4}, \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} \cdot u\right)}\right) \cdot -4 \]

   14. lower-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{4}, \color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{-1}{2} \cdot u\right)\right) \cdot -4 \]

   15. lower-*.f32 11.7

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

5. Applied rewrites 11.7%

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

7. Applied rewrites 11.7%

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

Alternative 9: 11.6% accurate, 36.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot -4 \]

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-out-- N/A

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

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{4}, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{-1}{2}}\right) \cdot u\right) \cdot -4 \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{4}, \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} \cdot u\right)}\right) \cdot -4 \]

   13. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{4}, \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} \cdot u\right)}\right) \cdot -4 \]

   14. lower-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{4}, \color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{-1}{2} \cdot u\right)\right) \cdot -4 \]

   15. lower-*.f32 11.7

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

5. Applied rewrites 11.7%

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

6. Taylor expanded in u around 0

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

8. Applied rewrites 11.7%

   \[\leadsto \mathsf{fma}\left(u \cdot \pi, \color{blue}{2}, -\pi\right) \]
9. Add Preprocessing

Alternative 10: 11.4% accurate, 170.0× 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 99.0%

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

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)} \]

   2. lower-neg.f32 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)} \]

   3. lower-PI.f32 11.5

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

5. Applied rewrites 11.5%

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

Alternative 11: 10.3% accurate, 510.0× 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 99.0%

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

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

7. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. distribute-rgt-neg-in N/A

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

   3. distribute-rgt-out N/A

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

   4. metadata-eval N/A

      \[\leadsto s \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{\frac{1}{2} + u \cdot \color{blue}{0}} - 1\right)\right)\right) \]

   5. mul0-rgt N/A

      \[\leadsto s \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{\frac{1}{2} + \color{blue}{0}} - 1\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto s \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{\color{blue}{\frac{1}{2}}} - 1\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto s \cdot \left(\mathsf{neg}\left(\log \left(\color{blue}{2} - 1\right)\right)\right) \]

   8. metadata-eval N/A

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

   9. metadata-eval N/A

      \[\leadsto s \cdot \left(\mathsf{neg}\left(\color{blue}{0}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto s \cdot \color{blue}{0} \]

   11. lower-*.f32 10.2

       \[\leadsto \color{blue}{s \cdot 0} \]

9. Applied rewrites 10.2%

   \[\leadsto \color{blue}{s \cdot 0} \]

10. Taylor expanded in s around 0

    \[\leadsto 0 \]
11. Step-by-step derivation

12. Applied rewrites 10.2%

    \[\leadsto 0 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(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))))