Harley's example

Percentage Accurate: 90.9% → 94.6%

Time: 48.5s

Alternatives: 3

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 3.018727915263448e-159
  2. c_n = 5.0811877166240356e-163
  3. t = -2.136133204032757e+29
  4. s = 2.072360005249832e-57

Specification

\[0 < c\_p \land 0 < c\_n\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := \frac{1}{1 + e^{-s}}\\ \frac{{t\_2}^{c\_p} \cdot {\left(1 - t\_2\right)}^{c\_n}}{{t\_1}^{c\_p} \cdot {\left(1 - t\_1\right)}^{c\_n}} \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ 1.0 (exp (- t))))) (t_2 (/ 1.0 (+ 1.0 (exp (- s))))))
   (/
    (* (pow t_2 c_p) (pow (- 1.0 t_2) c_n))
    (* (pow t_1 c_p) (pow (- 1.0 t_1) c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + exp(-t));
	double t_2 = 1.0 / (1.0 + exp(-s));
	return (pow(t_2, c_p) * pow((1.0 - t_2), c_n)) / (pow(t_1, c_p) * pow((1.0 - t_1), c_n));
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 1.0d0 / (1.0d0 + exp(-t))
    t_2 = 1.0d0 / (1.0d0 + exp(-s))
    code = ((t_2 ** c_p) * ((1.0d0 - t_2) ** c_n)) / ((t_1 ** c_p) * ((1.0d0 - t_1) ** c_n))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + Math.exp(-t));
	double t_2 = 1.0 / (1.0 + Math.exp(-s));
	return (Math.pow(t_2, c_p) * Math.pow((1.0 - t_2), c_n)) / (Math.pow(t_1, c_p) * Math.pow((1.0 - t_1), c_n));
}

Python

def code(c_p, c_n, t, s):
	t_1 = 1.0 / (1.0 + math.exp(-t))
	t_2 = 1.0 / (1.0 + math.exp(-s))
	return (math.pow(t_2, c_p) * math.pow((1.0 - t_2), c_n)) / (math.pow(t_1, c_p) * math.pow((1.0 - t_1), c_n))

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 / Float64(1.0 + exp(Float64(-t))))
	t_2 = Float64(1.0 / Float64(1.0 + exp(Float64(-s))))
	return Float64(Float64((t_2 ^ c_p) * (Float64(1.0 - t_2) ^ c_n)) / Float64((t_1 ^ c_p) * (Float64(1.0 - t_1) ^ c_n)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	t_1 = 1.0 / (1.0 + exp(-t));
	t_2 = 1.0 / (1.0 + exp(-s));
	tmp = ((t_2 ^ c_p) * ((1.0 - t_2) ^ c_n)) / ((t_1 ^ c_p) * ((1.0 - t_1) ^ c_n));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$2, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := \frac{1}{1 + e^{-s}}\\ \frac{{t\_2}^{c\_p} \cdot {\left(1 - t\_2\right)}^{c\_n}}{{t\_1}^{c\_p} \cdot {\left(1 - t\_1\right)}^{c\_n}} \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ 1.0 (exp (- t))))) (t_2 (/ 1.0 (+ 1.0 (exp (- s))))))
   (/
    (* (pow t_2 c_p) (pow (- 1.0 t_2) c_n))
    (* (pow t_1 c_p) (pow (- 1.0 t_1) c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + exp(-t));
	double t_2 = 1.0 / (1.0 + exp(-s));
	return (pow(t_2, c_p) * pow((1.0 - t_2), c_n)) / (pow(t_1, c_p) * pow((1.0 - t_1), c_n));
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 1.0d0 / (1.0d0 + exp(-t))
    t_2 = 1.0d0 / (1.0d0 + exp(-s))
    code = ((t_2 ** c_p) * ((1.0d0 - t_2) ** c_n)) / ((t_1 ** c_p) * ((1.0d0 - t_1) ** c_n))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + Math.exp(-t));
	double t_2 = 1.0 / (1.0 + Math.exp(-s));
	return (Math.pow(t_2, c_p) * Math.pow((1.0 - t_2), c_n)) / (Math.pow(t_1, c_p) * Math.pow((1.0 - t_1), c_n));
}

Python

def code(c_p, c_n, t, s):
	t_1 = 1.0 / (1.0 + math.exp(-t))
	t_2 = 1.0 / (1.0 + math.exp(-s))
	return (math.pow(t_2, c_p) * math.pow((1.0 - t_2), c_n)) / (math.pow(t_1, c_p) * math.pow((1.0 - t_1), c_n))

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 / Float64(1.0 + exp(Float64(-t))))
	t_2 = Float64(1.0 / Float64(1.0 + exp(Float64(-s))))
	return Float64(Float64((t_2 ^ c_p) * (Float64(1.0 - t_2) ^ c_n)) / Float64((t_1 ^ c_p) * (Float64(1.0 - t_1) ^ c_n)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	t_1 = 1.0 / (1.0 + exp(-t));
	t_2 = 1.0 / (1.0 + exp(-s));
	tmp = ((t_2 ^ c_p) * ((1.0 - t_2) ^ c_n)) / ((t_1 ^ c_p) * ((1.0 - t_1) ^ c_n));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$2, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 94.6% accurate, 896.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 1.0)

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = 1.0d0
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0;
}

Python

def code(c_p, c_n, t, s):
	return 1.0

Julia

function code(c_p, c_n, t, s)
	return 1.0
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := 1.0

Derivation

1. Initial program 92.6%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p} \cdot {\color{blue}{\frac{1}{2}}}^{c\_n}} \]
4. Step-by-step derivation

5. Simplified 93.0%

   \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\color{blue}{0.5}}^{c\_n}} \]

6. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(1 + e^{\mathsf{neg}\left(t\right)}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(t\right)} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot e^{\mathsf{neg}\left(t\right)} + \frac{1}{2} \cdot 1} \]

   3. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot e^{\mathsf{neg}\left(t\right)} + \color{blue}{\frac{1}{2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, e^{\mathsf{neg}\left(t\right)}, \frac{1}{2}\right)} \]

   5. neg-mul-1 N/A

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

   6. lower-exp.f64 N/A

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

   7. neg-mul-1 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, e^{\color{blue}{\mathsf{neg}\left(t\right)}}, \frac{1}{2}\right) \]

   8. lower-neg.f64 95.7

      \[\leadsto \mathsf{fma}\left(0.5, e^{\color{blue}{-t}}, 0.5\right) \]

8. Simplified 95.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, e^{-t}, 0.5\right)} \]

9. Taylor expanded in t around 0

   \[\leadsto \color{blue}{1} \]
10. Step-by-step derivation

11. Simplified 98.3%

    \[\leadsto \color{blue}{1} \]
12. Add Preprocessing

Alternative 2: 14.1% accurate, 896.0× speedup

Math

\[\begin{array}{l} \\ 0.015625 \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 0.015625)

C

double code(double c_p, double c_n, double t, double s) {
	return 0.015625;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = 0.015625d0
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 0.015625;
}

Python

def code(c_p, c_n, t, s):
	return 0.015625

Julia

function code(c_p, c_n, t, s)
	return 0.015625
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 0.015625;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := 0.015625

Derivation

1. Initial program 92.6%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{8 \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2}}{1 + e^{\mathsf{neg}\left(s\right)}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{8 \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2}}{1 + e^{\mathsf{neg}\left(s\right)}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot 8}}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot 8}}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2}} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   6. sub-neg N/A

      \[\leadsto \frac{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right)}}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right)}}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   8. distribute-neg-frac N/A

      \[\leadsto \frac{{\left(1 + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{{\left(1 + \frac{\color{blue}{-1}}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{{\left(1 + \color{blue}{\frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{\color{blue}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   12. lower-exp.f64 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + \color{blue}{e^{\mathsf{neg}\left(s\right)}}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   13. lower-neg.f64 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{\mathsf{neg}\left(s\right)}}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   14. lower-+.f64 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot 8}{\color{blue}{1 + e^{\mathsf{neg}\left(s\right)}}} \]

   15. lower-exp.f64 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot 8}{1 + \color{blue}{e^{\mathsf{neg}\left(s\right)}}} \]

5. Simplified 96.3%

   \[\leadsto \color{blue}{\frac{{\left(1 + \frac{-1}{1 + e^{-s}}\right)}^{2} \cdot 8}{1 + e^{-s}}} \]

6. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{1}{64}} \]
7. Step-by-step derivation

8. Simplified 14.5%

   \[\leadsto \color{blue}{0.015625} \]
9. Add Preprocessing

Alternative 3: 13.8% accurate, 896.0× speedup

Math

\[\begin{array}{l} \\ 0.0078125 \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 0.0078125)

C

double code(double c_p, double c_n, double t, double s) {
	return 0.0078125;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = 0.0078125d0
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 0.0078125;
}

Python

def code(c_p, c_n, t, s):
	return 0.0078125

Julia

function code(c_p, c_n, t, s)
	return 0.0078125
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 0.0078125;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := 0.0078125

Derivation

1. Initial program 92.6%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

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

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{8 \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2}}{1 + e^{\mathsf{neg}\left(s\right)}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{8 \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2}}{1 + e^{\mathsf{neg}\left(s\right)}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot 8}}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot 8}}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2}} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   6. sub-neg N/A

      \[\leadsto \frac{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right)}}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right)}}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   8. distribute-neg-frac N/A

      \[\leadsto \frac{{\left(1 + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{{\left(1 + \frac{\color{blue}{-1}}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{{\left(1 + \color{blue}{\frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{\color{blue}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   12. lower-exp.f64 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + \color{blue}{e^{\mathsf{neg}\left(s\right)}}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   13. lower-neg.f64 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{\mathsf{neg}\left(s\right)}}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   14. lower-+.f64 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot 8}{\color{blue}{1 + e^{\mathsf{neg}\left(s\right)}}} \]

   15. lower-exp.f64 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot 8}{1 + \color{blue}{e^{\mathsf{neg}\left(s\right)}}} \]

5. Simplified 96.3%

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

   1. lift-neg.f64 N/A

      \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{\mathsf{neg}\left(s\right)}}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   2. lift-exp.f64 N/A

      \[\leadsto \frac{{\left(1 + \frac{-1}{1 + \color{blue}{e^{\mathsf{neg}\left(s\right)}}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{{\left(1 + \frac{-1}{\color{blue}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{{\left(1 + \color{blue}{\frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{{\color{blue}{\left(1 + \frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{2} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(1 + \frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2}} \cdot 8}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(1 + \frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot 8}}{1 + e^{\mathsf{neg}\left(s\right)}} \]

   8. lift-neg.f64 N/A

      \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot 8}{1 + e^{\color{blue}{\mathsf{neg}\left(s\right)}}} \]

   9. lift-exp.f64 N/A

      \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot 8}{1 + \color{blue}{e^{\mathsf{neg}\left(s\right)}}} \]

   10. flip-+ N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot 8}{\color{blue}{\frac{1 \cdot 1 - e^{\mathsf{neg}\left(s\right)} \cdot e^{\mathsf{neg}\left(s\right)}}{1 - e^{\mathsf{neg}\left(s\right)}}}} \]

   11. associate-/r/ N/A

       \[\leadsto \color{blue}{\frac{{\left(1 + \frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{2} \cdot 8}{1 \cdot 1 - e^{\mathsf{neg}\left(s\right)} \cdot e^{\mathsf{neg}\left(s\right)}} \cdot \left(1 - e^{\mathsf{neg}\left(s\right)}\right)} \]

7. Applied egg-rr 4.7%

   \[\leadsto \color{blue}{\frac{{\left(1 + \frac{-1}{1 + e^{-s}}\right)}^{2} \cdot 8}{1 - e^{\left(-s\right) - s}} \cdot \left(1 - e^{-s}\right)} \]

8. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{1}{128}} \]
9. Step-by-step derivation

10. Simplified 14.2%

    \[\leadsto \color{blue}{0.0078125} \]
11. Add Preprocessing

Developer Target 1: 96.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c\_n} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (*
  (pow (/ (+ 1.0 (exp (- t))) (+ 1.0 (exp (- s)))) c_p)
  (pow (/ (+ 1.0 (exp t)) (+ 1.0 (exp s))) c_n)))

C

double code(double c_p, double c_n, double t, double s) {
	return pow(((1.0 + exp(-t)) / (1.0 + exp(-s))), c_p) * pow(((1.0 + exp(t)) / (1.0 + exp(s))), c_n);
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = (((1.0d0 + exp(-t)) / (1.0d0 + exp(-s))) ** c_p) * (((1.0d0 + exp(t)) / (1.0d0 + exp(s))) ** c_n)
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.pow(((1.0 + Math.exp(-t)) / (1.0 + Math.exp(-s))), c_p) * Math.pow(((1.0 + Math.exp(t)) / (1.0 + Math.exp(s))), c_n);
}

Python

def code(c_p, c_n, t, s):
	return math.pow(((1.0 + math.exp(-t)) / (1.0 + math.exp(-s))), c_p) * math.pow(((1.0 + math.exp(t)) / (1.0 + math.exp(s))), c_n)

Julia

function code(c_p, c_n, t, s)
	return Float64((Float64(Float64(1.0 + exp(Float64(-t))) / Float64(1.0 + exp(Float64(-s)))) ^ c_p) * (Float64(Float64(1.0 + exp(t)) / Float64(1.0 + exp(s))) ^ c_n))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = (((1.0 + exp(-t)) / (1.0 + exp(-s))) ^ c_p) * (((1.0 + exp(t)) / (1.0 + exp(s))) ^ c_n);
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(N[Power[N[(N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision] * N[Power[N[(N[(1.0 + N[Exp[t], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Exp[s], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024214 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0.0 c_p) (< 0.0 c_n))

  :alt
  (! :herbie-platform default (* (pow (/ (+ 1 (exp (- t))) (+ 1 (exp (- s)))) c_p) (pow (/ (+ 1 (exp t)) (+ 1 (exp s))) c_n)))

  (/ (* (pow (/ 1.0 (+ 1.0 (exp (- s)))) c_p) (pow (- 1.0 (/ 1.0 (+ 1.0 (exp (- s))))) c_n)) (* (pow (/ 1.0 (+ 1.0 (exp (- t)))) c_p) (pow (- 1.0 (/ 1.0 (+ 1.0 (exp (- t))))) c_n))))