Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 15.1s

Alternatives: 9

Speedup: 1.4×

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, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = log(((single(1.0) / ((u / (single(1.0) + exp(-(single(pi) / s)))) + ((single(1.0) - u) / (single(1.0) + exp((single(pi) / s)))))) + single(-1.0))) * -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. Step-by-step derivation

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 2: 37.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = log((single(-1.0) + (single(1.0) / ((u * single(0.5)) + ((single(1.0) - u) / (single(1.0) + exp((single(-1.0) + (single(1.0) + (single(pi) / s)))))))))) * -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. Step-by-step derivation

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 37.7%

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

   1. expm1-log1p-u 37.7%

      \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + 1} + \frac{1 - u}{1 + e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{s}\right)\right)}}}} + -1\right)\right) \]

6. Applied egg-rr 37.7%

   \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + 1} + \frac{1 - u}{1 + e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\pi}{s}\right)\right)}}}} + -1\right)\right) \]
7. Step-by-step derivation

   1. expm1-udef 37.7%

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

   2. log1p-udef 37.7%

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

   3. add-exp-log 37.7%

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

8. Applied egg-rr 37.7%

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

   1. distribute-rgt-neg-out 37.7%

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

   2. div-inv 37.7%

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

   3. metadata-eval 37.7%

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

   4. metadata-eval 37.7%

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

   5. sub-neg 37.7%

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

   6. metadata-eval 37.7%

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

10. Applied egg-rr 37.7%

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

11. Final simplification 37.7%

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

Alternative 3: 37.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = log((single(-1.0) + (single(1.0) / (((single(1.0) - u) / (single(1.0) + exp((single(pi) / s)))) + (u * single(0.5)))))) * -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. Step-by-step derivation

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 37.7%

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

   1. distribute-rgt-neg-out 37.7%

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

   2. div-inv 37.7%

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

   3. metadata-eval 37.7%

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

   4. metadata-eval 37.7%

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

   5. +-commutative 37.7%

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

6. Applied egg-rr 37.7%

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

7. Final simplification 37.7%

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

Alternative 4: 37.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * -exp(log(log(((-1.0e0) + (2.0e0 / u)))))
end function

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * -exp(log(log((single(-1.0) + (single(2.0) / u)))));
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. Step-by-step derivation

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 37.7%

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

5. Taylor expanded in s around inf 37.0%

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

   1. associate-+r+ 37.0%

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

   2. unpow2 37.0%

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

   3. unpow2 37.0%

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

   4. times-frac 37.0%

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

   5. unpow1 37.0%

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

   6. pow-plus 37.0%

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

   7. metadata-eval 37.0%

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

7. Simplified 37.0%

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

8. Taylor expanded in s around 0 37.3%

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

   1. associate-*r* 37.3%

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

   2. neg-mul-1 37.3%

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

   3. sub-neg 37.3%

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

   4. associate-*r/ 37.3%

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

   5. metadata-eval 37.3%

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

   6. metadata-eval 37.3%

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

10. Simplified 37.3%

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

    1. add-exp-log 37.3%

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

    2. +-commutative 37.3%

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

12. Applied egg-rr 37.3%

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

13. Final simplification 37.3%

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

Alternative 5: 37.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(\frac{-1 + {\left(\frac{2}{u}\right)}^{3}}{\frac{2}{u} \cdot \frac{2}{u} + \left(1 + \frac{2}{u}\right)}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (/
     (+ -1.0 (pow (/ 2.0 u) 3.0))
     (+ (* (/ 2.0 u) (/ 2.0 u)) (+ 1.0 (/ 2.0 u))))))))

C

float code(float u, float s) {
	return s * -logf(((-1.0f + powf((2.0f / u), 3.0f)) / (((2.0f / u) * (2.0f / u)) + (1.0f + (2.0f / u)))));
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * -log((((-1.0e0) + ((2.0e0 / u) ** 3.0e0)) / (((2.0e0 / u) * (2.0e0 / u)) + (1.0e0 + (2.0e0 / u)))))
end function

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(Float32(-1.0) + (Float32(Float32(2.0) / u) ^ Float32(3.0))) / Float32(Float32(Float32(Float32(2.0) / u) * Float32(Float32(2.0) / u)) + Float32(Float32(1.0) + Float32(Float32(2.0) / u)))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log(((single(-1.0) + ((single(2.0) / u) ^ single(3.0))) / (((single(2.0) / u) * (single(2.0) / u)) + (single(1.0) + (single(2.0) / u)))));
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. Step-by-step derivation

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 37.7%

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

5. Taylor expanded in s around inf 37.0%

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

   1. associate-+r+ 37.0%

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

   2. unpow2 37.0%

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

   3. unpow2 37.0%

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

   4. times-frac 37.0%

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

   5. unpow1 37.0%

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

   6. pow-plus 37.0%

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

   7. metadata-eval 37.0%

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

7. Simplified 37.0%

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

8. Taylor expanded in s around 0 37.3%

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

   1. associate-*r* 37.3%

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

   2. neg-mul-1 37.3%

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

   3. sub-neg 37.3%

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

   4. associate-*r/ 37.3%

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

   5. metadata-eval 37.3%

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

   6. metadata-eval 37.3%

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

10. Simplified 37.3%

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

    1. flip3-+ 37.3%

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

    2. metadata-eval 37.3%

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

    3. metadata-eval 37.3%

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

12. Applied egg-rr 37.3%

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

13. Final simplification 37.3%

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

Alternative 6: 37.1% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * -log((((-1.0e0) + ((2.0e0 / u) * (2.0e0 / u))) / ((2.0e0 / u) - (-1.0e0))))
end function

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * -log(((single(-1.0) + ((single(2.0) / u) * (single(2.0) / u))) / ((single(2.0) / u) - single(-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. Step-by-step derivation

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 37.7%

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

5. Taylor expanded in s around inf 37.0%

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

   1. associate-+r+ 37.0%

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

   2. unpow2 37.0%

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

   3. unpow2 37.0%

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

   4. times-frac 37.0%

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

   5. unpow1 37.0%

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

   6. pow-plus 37.0%

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

   7. metadata-eval 37.0%

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

7. Simplified 37.0%

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

8. Taylor expanded in s around 0 37.3%

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

   1. associate-*r* 37.3%

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

   2. neg-mul-1 37.3%

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

   3. sub-neg 37.3%

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

   4. associate-*r/ 37.3%

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

   5. metadata-eval 37.3%

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

   6. metadata-eval 37.3%

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

10. Simplified 37.3%

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

    1. flip-+ 37.3%

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

    2. metadata-eval 37.3%

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

12. Applied egg-rr 37.3%

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

13. Final simplification 37.3%

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

Alternative 7: 37.1% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * -log(((-1.0e0) + (2.0e0 / u)))
end function

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(-1.0) + (single(2.0) / u)));
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. Step-by-step derivation

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 37.7%

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

5. Taylor expanded in s around inf 37.0%

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

   1. associate-+r+ 37.0%

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

   2. unpow2 37.0%

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

   3. unpow2 37.0%

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

   4. times-frac 37.0%

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

   5. unpow1 37.0%

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

   6. pow-plus 37.0%

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

   7. metadata-eval 37.0%

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

7. Simplified 37.0%

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

8. Taylor expanded in s around 0 37.3%

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

   1. associate-*r* 37.3%

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

   2. neg-mul-1 37.3%

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

   3. sub-neg 37.3%

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

   4. associate-*r/ 37.3%

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

   5. metadata-eval 37.3%

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

   6. metadata-eval 37.3%

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

10. Simplified 37.3%

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

11. Final simplification 37.3%

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

Alternative 8: 37.0% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = -s * log((2.0e0 / u))
end function

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(2.0) / u));
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. Step-by-step derivation

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 37.7%

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

5. Taylor expanded in s around inf 37.0%

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

   1. associate-+r+ 37.0%

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

   2. unpow2 37.0%

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

   3. unpow2 37.0%

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

   4. times-frac 37.0%

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

   5. unpow1 37.0%

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

   6. pow-plus 37.0%

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

   7. metadata-eval 37.0%

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

7. Simplified 37.0%

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

8. Taylor expanded in s around 0 37.3%

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

   1. associate-*r* 37.3%

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

   2. neg-mul-1 37.3%

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

   3. sub-neg 37.3%

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

   4. associate-*r/ 37.3%

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

   5. metadata-eval 37.3%

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

   6. metadata-eval 37.3%

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

10. Simplified 37.3%

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

11. Taylor expanded in u around 0 37.2%

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

12. Final simplification 37.2%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{2}{u}\right) \]

Alternative 9: 11.4% 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 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. Step-by-step derivation

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u around 0 11.5%

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

   1. mul-1-neg 11.5%

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

6. Simplified 11.5%

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

7. Final simplification 11.5%

   \[\leadsto -\pi \]

Reproduce

herbie shell --seed 2023207 
(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))))