Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 7.6s

Alternatives: 15

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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 99.0%

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

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ t_1 := \frac{1}{t\_0 + 1}\\ t_2 := \frac{1}{1 + t\_0}\\ t_3 := \mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - t\_2, t\_2\right)\\ \left(-s\right) \cdot \log \left(\frac{{\left(\mathsf{fma}\left(\frac{1}{e^{\frac{-\pi}{s}} + 1} - t\_1, u, t\_1\right)\right)}^{-3} - 1}{1 + \left(\frac{1}{t\_3} + {t\_3}^{-2}\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s)))
        (t_1 (/ 1.0 (+ t_0 1.0)))
        (t_2 (/ 1.0 (+ 1.0 t_0)))
        (t_3 (fma u (- (/ 1.0 (+ 1.0 (exp (* -1.0 (/ PI s))))) t_2) t_2)))
   (*
    (- s)
    (log
     (/
      (- (pow (fma (- (/ 1.0 (+ (exp (/ (- PI) s)) 1.0)) t_1) u t_1) -3.0) 1.0)
      (+ 1.0 (+ (/ 1.0 t_3) (pow t_3 -2.0))))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	float t_1 = 1.0f / (t_0 + 1.0f);
	float t_2 = 1.0f / (1.0f + t_0);
	float t_3 = fmaf(u, ((1.0f / (1.0f + expf((-1.0f * (((float) M_PI) / s))))) - t_2), t_2);
	return -s * logf(((powf(fmaf(((1.0f / (expf((-((float) M_PI) / s)) + 1.0f)) - t_1), u, t_1), -3.0f) - 1.0f) / (1.0f + ((1.0f / t_3) + powf(t_3, -2.0f)))));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	t_1 = Float32(Float32(1.0) / Float32(t_0 + Float32(1.0)))
	t_2 = Float32(Float32(1.0) / Float32(Float32(1.0) + t_0))
	t_3 = fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-1.0) * Float32(Float32(pi) / s))))) - t_2), t_2)
	return Float32(Float32(-s) * log(Float32(Float32((fma(Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(-Float32(pi)) / s)) + Float32(1.0))) - t_1), u, t_1) ^ Float32(-3.0)) - Float32(1.0)) / Float32(Float32(1.0) + Float32(Float32(Float32(1.0) / t_3) + (t_3 ^ Float32(-2.0)))))))
end

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 99.0%

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

4. Taylor expanded in s around 0

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

6. Applied rewrites 99.0%

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

Alternative 3: 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

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 99.0%

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

Alternative 5: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\left(\log \left(\frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + -1 \cdot \left(-\log u\right)\right) \cdot -1\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (expm1
    (*
     (+
      (log
       (-
        (/ 1.0 (+ 1.0 (exp (* -1.0 (/ PI s)))))
        (/ 1.0 (+ 1.0 (exp (/ PI s))))))
      (* -1.0 (- (log u))))
     -1.0)))))

C

float code(float u, float s) {
	return -s * logf(expm1f(((logf(((1.0f / (1.0f + expf((-1.0f * (((float) M_PI) / s))))) - (1.0f / (1.0f + expf((((float) M_PI) / s)))))) + (-1.0f * -logf(u))) * -1.0f)));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(expm1(Float32(Float32(log(Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-1.0) * Float32(Float32(pi) / s))))) - Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))) + Float32(Float32(-1.0) * Float32(-log(u)))) * Float32(-1.0)))))
end

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in u around inf

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

   1. lower-+.f32 N/A

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

6. Applied rewrites 97.7%

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

Alternative 6: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right) \cdot -1\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (expm1
    (*
     (log
      (*
       u
       (-
        (/ 1.0 (+ 1.0 (exp (* -1.0 (/ PI s)))))
        (/ 1.0 (+ 1.0 (exp (/ PI s)))))))
     -1.0)))))

C

float code(float u, float s) {
	return -s * logf(expm1f((logf((u * ((1.0f / (1.0f + expf((-1.0f * (((float) M_PI) / s))))) - (1.0f / (1.0f + expf((((float) M_PI) / s))))))) * -1.0f)));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(expm1(Float32(log(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-1.0) * Float32(Float32(pi) / s))))) - Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))))) * Float32(-1.0)))))
end

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in u around inf

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-+.f32 N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-/.f32 N/A

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

6. Applied rewrites 97.7%

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

Alternative 7: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(-Float32(pi)) / s)) + Float32(1.0))) - Float32(Float32(1.0) / Float32(exp(Float32(Float32(pi) / s)) + Float32(1.0)))) * u)) - Float32(1.0))))
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) / (exp((single(pi) / s)) + single(1.0)))) * 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. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 97.8%

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

Alternative 8: 96.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf(((1.0f / (((1.0f / (expf((-((float) M_PI) / s)) + 1.0f)) - (1.0f / (2.0f + (0.5f * ((((float) M_PI) * ((float) M_PI)) / (s * s)))))) * u)) - 1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(-Float32(pi)) / s)) + Float32(1.0))) - Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(0.5) * Float32(Float32(Float32(pi) * Float32(pi)) / Float32(s * s)))))) * u)) - Float32(1.0))))
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(2.0) + (single(0.5) * ((single(pi) * single(pi)) / (s * s)))))) * 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. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 97.8%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-PI.f32 96.7

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in s around 0

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

   1. pow2 N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lower-*.f32 96.7

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

10. Applied rewrites 96.7%

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

Alternative 9: 96.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(-Float32(pi)) / s)) + Float32(1.0))) - Float32(Float32(1.0) / Float32(Float32(0.5) * Float32(Float32(Float32(pi) * Float32(pi)) / Float32(s * s))))) * u)) - Float32(1.0))))
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(0.5) * ((single(pi) * single(pi)) / (s * s))))) * 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. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 97.8%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-PI.f32 96.7

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in s around 0

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

   1. pow2 N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lower-*.f32 96.7

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

10. Applied rewrites 96.7%

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

Alternative 10: 94.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(1.0) / Float32(exp(Float32(Float32(-Float32(pi)) / s)) + Float32(1.0))) - Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(pi) / s)))) * u)) - Float32(1.0))))
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(2.0) + (single(pi) / s)))) * 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. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 97.8%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 94.6

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

7. Applied rewrites 94.6%

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

Alternative 11: 37.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / ((single(0.5) - (single(1.0) / (exp((single(pi) / s)) + single(1.0)))) * 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. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 97.8%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 37.2%

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

Alternative 12: 37.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf(((1.0f / ((0.5f - (1.0f / (2.0f + (0.5f * ((((float) M_PI) * ((float) M_PI)) / (s * s)))))) * u)) - 1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(0.5) - Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(0.5) * Float32(Float32(Float32(pi) * Float32(pi)) / Float32(s * s)))))) * u)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / ((single(0.5) - (single(1.0) / (single(2.0) + (single(0.5) * ((single(pi) * single(pi)) / (s * s)))))) * 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. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 97.8%

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

5. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-PI.f32 96.7

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in s around 0

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

   1. pow2 N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lower-*.f32 96.7

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

10. Applied rewrites 96.7%

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

11. Taylor expanded in s around inf

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

13. Applied rewrites 37.2%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\left(0.5 - \frac{1}{2 + 0.5 \cdot \frac{\pi \cdot \pi}{s \cdot s}}\right) \cdot u} - 1\right) \]
14. Add Preprocessing

Alternative 13: 25.1% 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 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. 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. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 24.9%

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

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. lift-/.f32 N/A

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

   3. lift-PI.f32 25.1

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

7. Applied rewrites 25.1%

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

Alternative 14: 11.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(fma(Float32(Float32(pi) * Float32(0.5)), u, Float32(Float32(-0.25) * Float32(pi))) * 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. 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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 11.5%

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

Alternative 15: 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 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. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lift-neg.f32 N/A

      \[\leadsto -\mathsf{PI}\left(\right) \]

   3. lift-PI.f32 11.3

      \[\leadsto -\pi \]

4. Applied rewrites 11.3%

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

Reproduce

herbie shell --seed 2025122 
(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))))