Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 18.0s

Alternatives: 14

Speedup: 1.0×

Specification

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 99.0%

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

8. Applied rewrites 98.8%

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

10. Applied rewrites 99.0%

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

11. Final simplification 99.0%

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

Alternative 2: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s));
	t_1 = (u / (single(1.0) + exp(-(single(pi) / s)))) + ((u / (single(-1.0) - t_0)) + (single(1.0) / (single(1.0) + t_0)));
	tmp = -s * log(((single(1.0) - (t_1 ^ single(-2.0))) / (single(-1.0) + (single(-1.0) / t_1))));
end

Derivation

1. Initial program 98.9%

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 99.0%

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

8. Applied rewrites 98.8%

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

10. Applied rewrites 99.0%

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

11. Final simplification 99.0%

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

Alternative 3: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 99.0%

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

8. Applied rewrites 98.9%

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

9. Final simplification 98.9%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 99.0%

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

8. Applied rewrites 98.8%

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

10. Applied rewrites 98.9%

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

11. Final simplification 98.9%

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

Alternative 5: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 99.0%

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

8. Applied rewrites 98.9%

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

9. Final simplification 98.9%

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

Alternative 6: 97.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u around inf

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-exp.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. mul-1-neg N/A

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

   10. lower-/.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f32 N/A

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

   14. distribute-neg-frac N/A

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

5. Applied rewrites 98.4%

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

6. Final simplification 98.4%

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

Alternative 7: 25.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 24.5%

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

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-log1p.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{fma}\left(-2, u \cdot \frac{\mathsf{PI}\left(\right)}{s \cdot \left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}, \color{blue}{\mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}\right) \]

   12. lower-/.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{fma}\left(-2, u \cdot \frac{\mathsf{PI}\left(\right)}{s \cdot \left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}, \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right) \]

   13. lower-PI.f32 24.7

       \[\leadsto \left(-s\right) \cdot \mathsf{fma}\left(-2, u \cdot \frac{\pi}{s \cdot \left(1 + \frac{\pi}{s}\right)}, \mathsf{log1p}\left(\frac{\color{blue}{\pi}}{s}\right)\right) \]

8. Applied rewrites 24.7%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\mathsf{fma}\left(-2, u \cdot \frac{\pi}{s \cdot \left(1 + \frac{\pi}{s}\right)}, \mathsf{log1p}\left(\frac{\pi}{s}\right)\right)} \]

9. Taylor expanded in s around 0

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

    1. lower-+.f32 N/A

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

    2. lower-log.f32 N/A

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

    3. lower-PI.f32 N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f32 N/A

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

    6. mul-1-neg N/A

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

    7. lower-neg.f32 N/A

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

    8. lower-log.f32 25.1

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

11. Applied rewrites 25.1%

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

12. Final simplification 25.1%

    \[\leadsto s \cdot \left(\left(-\log \pi\right) - \mathsf{fma}\left(u, -2, -\log s\right)\right) \]
13. Add Preprocessing

Alternative 8: 25.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{\pi}{s}\\ \mathsf{fma}\left(u, 2 \cdot \mathsf{fma}\left(u, \frac{\pi \cdot \pi}{s \cdot \left(t\_0 \cdot t\_0\right)}, \frac{\pi}{t\_0}\right), \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (/ PI s))))
   (fma
    u
    (* 2.0 (fma u (/ (* PI PI) (* s (* t_0 t_0))) (/ PI t_0)))
    (* (- s) (log1p (/ PI s))))))

C

float code(float u, float s) {
	float t_0 = 1.0f + (((float) M_PI) / s);
	return fmaf(u, (2.0f * fmaf(u, ((((float) M_PI) * ((float) M_PI)) / (s * (t_0 * t_0))), (((float) M_PI) / t_0))), (-s * log1pf((((float) M_PI) / s))));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) + Float32(Float32(pi) / s))
	return fma(u, Float32(Float32(2.0) * fma(u, Float32(Float32(Float32(pi) * Float32(pi)) / Float32(s * Float32(t_0 * t_0))), Float32(Float32(pi) / t_0))), Float32(Float32(-s) * log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 24.5%

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

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

8. Applied rewrites 24.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(u, 2 \cdot \mathsf{fma}\left(u, \frac{\pi \cdot \pi}{s \cdot \left(\left(1 + \frac{\pi}{s}\right) \cdot \left(1 + \frac{\pi}{s}\right)\right)}, \frac{\pi}{1 + \frac{\pi}{s}}\right), \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)\right)} \]
9. Add Preprocessing

Alternative 9: 25.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \mathsf{fma}\left(-2, u \cdot \frac{\pi}{\pi}, \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (fma -2.0 (* u (/ PI PI)) (log1p (/ PI s)))))

C

float code(float u, float s) {
	return -s * fmaf(-2.0f, (u * (((float) M_PI) / ((float) M_PI))), log1pf((((float) M_PI) / s)));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * fma(Float32(-2.0), Float32(u * Float32(Float32(pi) / Float32(pi))), log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 24.5%

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

6. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-log1p.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{fma}\left(-2, u \cdot \frac{\mathsf{PI}\left(\right)}{s \cdot \left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}, \color{blue}{\mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)}\right) \]

   12. lower-/.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{fma}\left(-2, u \cdot \frac{\mathsf{PI}\left(\right)}{s \cdot \left(1 + \frac{\mathsf{PI}\left(\right)}{s}\right)}, \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right)\right) \]

   13. lower-PI.f32 24.7

       \[\leadsto \left(-s\right) \cdot \mathsf{fma}\left(-2, u \cdot \frac{\pi}{s \cdot \left(1 + \frac{\pi}{s}\right)}, \mathsf{log1p}\left(\frac{\color{blue}{\pi}}{s}\right)\right) \]

8. Applied rewrites 24.7%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\mathsf{fma}\left(-2, u \cdot \frac{\pi}{s \cdot \left(1 + \frac{\pi}{s}\right)}, \mathsf{log1p}\left(\frac{\pi}{s}\right)\right)} \]

9. Taylor expanded in s around 0

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{fma}\left(-2, u \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{PI}\left(\right)}}, \mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)\right) \]
10. Step-by-step derivation

    1. lower-PI.f32 24.8

       \[\leadsto \left(-s\right) \cdot \mathsf{fma}\left(-2, u \cdot \frac{\pi}{\color{blue}{\pi}}, \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \]

11. Applied rewrites 24.8%

    \[\leadsto \left(-s\right) \cdot \mathsf{fma}\left(-2, u \cdot \frac{\pi}{\color{blue}{\pi}}, \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \]
12. Add Preprocessing

Alternative 10: 25.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (- s) (log1p (/ PI s))))

C

float code(float u, float s) {
	return -s * log1pf((((float) M_PI) / s));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log1p(Float32(Float32(pi) / s)))
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 24.5%

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

6. Taylor expanded in u around 0

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

   1. lower-log1p.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{\mathsf{log1p}\left(\frac{\mathsf{PI}\left(\right)}{s}\right)} \]

   2. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right) \]

   3. lower-PI.f32 24.7

      \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\color{blue}{\pi}}{s}\right) \]

8. Applied rewrites 24.7%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]
9. Add Preprocessing

Alternative 11: 11.8% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 24.5%

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

6. Taylor expanded in s around inf

   \[\leadsto \color{blue}{4 \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-fma.f32 11.0

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

8. Applied rewrites 11.0%

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

9. Taylor expanded in u around inf

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

    1. lower-*.f32 N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. lower-fma.f32 N/A

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

    5. lower-PI.f32 N/A

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

    6. mul-1-neg N/A

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

    7. lower-neg.f32 N/A

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

    8. lower-/.f32 N/A

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

    9. lower-PI.f32 11.0

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

11. Applied rewrites 11.0%

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

Alternative 12: 11.8% accurate, 36.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 24.5%

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

6. Taylor expanded in s around inf

   \[\leadsto \color{blue}{4 \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-fma.f32 11.0

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

8. Applied rewrites 11.0%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

       \[\leadsto \color{blue}{2 \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \mathsf{PI}\left(\right)} \]

    2. lower-fma.f32 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f32 N/A

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

    5. lower-PI.f32 N/A

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

    6. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot u, \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}\right) \]

    7. lower-neg.f32 N/A

       \[\leadsto \mathsf{fma}\left(2, \mathsf{PI}\left(\right) \cdot u, \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}\right) \]

    8. lower-PI.f32 11.0

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

11. Applied rewrites 11.0%

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

12. Final simplification 11.0%

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

Alternative 13: 11.6% accurate, 170.0× speedup

Math

\[\begin{array}{l} \\ -\pi \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (- PI))

C

float code(float u, float s) {
	return -((float) M_PI);
}

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -single(pi);
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-PI.f32 10.7

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

5. Applied rewrites 10.7%

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

Alternative 14: 10.3% accurate, 510.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 0.0)

C

float code(float u, float s) {
	return 0.0f;
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

function code(u, s)
	return Float32(0.0)
end

MATLAB

function tmp = code(u, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 10.2%

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

   1. lift-neg.f32 N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 10.2

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

7. Applied rewrites 10.2%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))