Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 9.4s

Alternatives: 10

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: 98.9% 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

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. Add Preprocessing

Alternative 2: 97.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in u around inf

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

   1. lower--.f32 N/A

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

4. Applied rewrites 97.7%

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

   1. lift--.f32 N/A

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

   2. sub-flip N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. add-to-fraction N/A

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

6. Applied rewrites 97.8%

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

   1. lift-fma.f32 N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. sub-flip-reverse N/A

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

   5. lower--.f32 97.8

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

8. Applied rewrites 97.8%

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

Alternative 3: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in u around inf

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

   1. lower--.f32 N/A

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 97.7%

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

Alternative 4: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower--.f32 N/A

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 97.7%

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

Alternative 5: 94.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in u around inf

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

   1. lower--.f32 N/A

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

4. Applied rewrites 97.7%

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

   1. lift--.f32 N/A

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

   2. sub-flip N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. add-to-fraction N/A

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

6. Applied rewrites 97.8%

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

   1. lift-fma.f32 N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. sub-flip-reverse N/A

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

   5. lower--.f32 97.8

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

8. Applied rewrites 97.8%

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

9. Taylor expanded in s around inf

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

    1. lower-+.f32 N/A

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

    2. lower-/.f32 N/A

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

    3. lower-PI.f32 94.5

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

11. Applied rewrites 94.5%

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

Alternative 6: 37.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(u * Float32(Float32(0.5) - Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))))) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / (u * (single(0.5) - (single(1.0) / (single(1.0) + exp((single(pi) / s))))))) - single(1.0)));
end

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in u around inf

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

   1. lower--.f32 N/A

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

4. Applied rewrites 97.7%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 37.2%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(0.5 - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} - 1\right) \]
8. Add Preprocessing

Alternative 7: 25.0% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around -inf

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

4. Applied rewrites 24.8%

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

5. Taylor expanded in u around 0

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right) \]

   3. lower-PI.f32 25.0

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

7. Applied rewrites 25.0%

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

Alternative 8: 11.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 11.5

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

4. Applied rewrites 11.5%

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

6. Applied rewrites 11.5%

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

Alternative 9: 11.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 11.5

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

4. Applied rewrites 11.5%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

   4. lift-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

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

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f32 N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval 11.5

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

6. Applied rewrites 11.5%

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

Alternative 10: 11.3% accurate, 46.3× 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%

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

2. Taylor expanded in u around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 11.3

      \[\leadsto -1 \cdot \pi \]

4. Applied rewrites 11.3%

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

   1. lift-*.f32 N/A

      \[\leadsto -1 \cdot \color{blue}{\pi} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\pi\right) \]

   3. lower-neg.f32 11.3

      \[\leadsto -\pi \]

6. Applied rewrites 11.3%

   \[\leadsto -\pi \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(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))))