Harley's example

Percentage Accurate: 91.0% → 99.7%

Time: 1.3min

Alternatives: 7

Speedup: 835.0×

• could not determine a ground truth

  1. c_p = 2.1586330391987695e-152
  2. c_n = 1.0474809381994848e-52
  3. t = 1.0215084980506294e+93
  4. s = 1.7061487735577814e+70

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: 91.0% 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: 99.7% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ e^{\left(t \cdot 0.5\right) \cdot \left(c\_n - c\_p\right) + s \cdot \left(\left(c\_n - c\_p\right) \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp (+ (* (* t 0.5) (- c_n c_p)) (* s (* (- c_n c_p) -0.5)))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp((((t * 0.5) * (c_n - c_p)) + (s * ((c_n - c_p) * -0.5))));
}

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 = exp((((t * 0.5d0) * (c_n - c_p)) + (s * ((c_n - c_p) * (-0.5d0)))))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp((((t * 0.5) * (c_n - c_p)) + (s * ((c_n - c_p) * -0.5))));
}

Python

def code(c_p, c_n, t, s):
	return math.exp((((t * 0.5) * (c_n - c_p)) + (s * ((c_n - c_p) * -0.5))))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(Float64(Float64(t * 0.5) * Float64(c_n - c_p)) + Float64(s * Float64(Float64(c_n - c_p) * -0.5))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = exp((((t * 0.5) * (c_n - c_p)) + (s * ((c_n - c_p) * -0.5))));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(N[(N[(t * 0.5), $MachinePrecision] * N[(c$95$n - c$95$p), $MachinePrecision]), $MachinePrecision] + N[(s * N[(N[(c$95$n - c$95$p), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 91.1%

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

4. Applied egg-rr 96.8%

   \[\leadsto \color{blue}{e^{c\_p \cdot \left(\left(0 - \mathsf{log1p}\left(\frac{1}{e^{s}}\right)\right) - \left(0 - \mathsf{log1p}\left(e^{0 - t}\right)\right)\right) + c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{s}}}\right) - \mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{t}}}\right)\right)}} \]

5. Taylor expanded in t around 0

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(c\_n \cdot \left(\log \left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right) - \log \frac{1}{2}\right) + \left(c\_p \cdot \left(\log 2 - \log \left(1 + \frac{1}{e^{s}}\right)\right) + t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(c\_n \cdot \left(\log \left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right) - \log \frac{1}{2}\right)\right), \left(c\_p \cdot \left(\log 2 - \log \left(1 + \frac{1}{e^{s}}\right)\right) + t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)\right)\right) \]

7. Simplified 99.6%

   \[\leadsto e^{\color{blue}{c\_n \cdot \left(\mathsf{log1p}\left(\frac{-1}{1 + e^{0 - s}}\right) - \log 0.5\right) + \left(c\_p \cdot \left(\log 2 - \mathsf{log1p}\left(e^{0 - s}\right)\right) + t \cdot \left(c\_p \cdot -0.5 + c\_n \cdot 0.5\right)\right)}} \]

8. Taylor expanded in s around 0

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right) + t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)}\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right) + s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_n + \frac{-1}{2} \cdot c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_n + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   5. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_n - \frac{1}{2} \cdot c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot \left(c\_n - c\_p\right)\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\left(t \cdot \frac{1}{2}\right) \cdot \left(c\_n - c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\left(\frac{1}{2} \cdot t\right) \cdot \left(c\_n - c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot t\right), \left(c\_n - c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \frac{1}{2}\right), \left(c\_n - c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \left(c\_n - c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(\left(-1 \cdot \frac{1}{2}\right) \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(-1 \cdot \left(\frac{1}{2} \cdot c\_n\right) + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(-1 \cdot \left(\frac{1}{2} \cdot c\_n\right) + \left(-1 \cdot \frac{-1}{2}\right) \cdot c\_p\right)\right)\right)\right) \]

   17. associate-*r* N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(-1 \cdot \left(\frac{1}{2} \cdot c\_n\right) + -1 \cdot \left(\frac{-1}{2} \cdot c\_p\right)\right)\right)\right)\right) \]

   18. distribute-lft-in N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(-1 \cdot \left(\frac{1}{2} \cdot c\_n + \frac{-1}{2} \cdot c\_p\right)\right)\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(-1 \cdot \left(\frac{1}{2} \cdot c\_n + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c\_p\right)\right)\right)\right)\right) \]

   20. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(-1 \cdot \left(\frac{1}{2} \cdot c\_n - \frac{1}{2} \cdot c\_p\right)\right)\right)\right)\right) \]

   21. distribute-lft-out-- N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(-1 \cdot \left(\frac{1}{2} \cdot \left(c\_n - c\_p\right)\right)\right)\right)\right)\right) \]

   22. associate-*r* N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(\left(-1 \cdot \frac{1}{2}\right) \cdot \left(c\_n - c\_p\right)\right)\right)\right)\right) \]

   23. metadata-eval N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(\frac{-1}{2} \cdot \left(c\_n - c\_p\right)\right)\right)\right)\right) \]

10. Simplified 100.0%

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

11. Final simplification 100.0%

    \[\leadsto e^{\left(t \cdot 0.5\right) \cdot \left(c\_n - c\_p\right) + s \cdot \left(\left(c\_n - c\_p\right) \cdot -0.5\right)} \]
12. Add Preprocessing

Alternative 2: 98.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ e^{-0.5 \cdot \left(\left(c\_n - c\_p\right) \cdot s\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 (exp (* -0.5 (* (- c_n c_p) s))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp((-0.5 * ((c_n - c_p) * s)));
}

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 = exp(((-0.5d0) * ((c_n - c_p) * s)))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp((-0.5 * ((c_n - c_p) * s)));
}

Python

def code(c_p, c_n, t, s):
	return math.exp((-0.5 * ((c_n - c_p) * s)))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(-0.5 * Float64(Float64(c_n - c_p) * s)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = exp((-0.5 * ((c_n - c_p) * s)));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(-0.5 * N[(N[(c$95$n - c$95$p), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 91.1%

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

4. Applied egg-rr 96.8%

   \[\leadsto \color{blue}{e^{c\_p \cdot \left(\left(0 - \mathsf{log1p}\left(\frac{1}{e^{s}}\right)\right) - \left(0 - \mathsf{log1p}\left(e^{0 - t}\right)\right)\right) + c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{s}}}\right) - \mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{t}}}\right)\right)}} \]

5. Taylor expanded in t around 0

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(c\_n \cdot \left(\log \left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right) - \log \frac{1}{2}\right) + \left(c\_p \cdot \left(\log 2 - \log \left(1 + \frac{1}{e^{s}}\right)\right) + t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(c\_n \cdot \left(\log \left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right) - \log \frac{1}{2}\right)\right), \left(c\_p \cdot \left(\log 2 - \log \left(1 + \frac{1}{e^{s}}\right)\right) + t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)\right)\right) \]

7. Simplified 99.6%

   \[\leadsto e^{\color{blue}{c\_n \cdot \left(\mathsf{log1p}\left(\frac{-1}{1 + e^{0 - s}}\right) - \log 0.5\right) + \left(c\_p \cdot \left(\log 2 - \mathsf{log1p}\left(e^{0 - s}\right)\right) + t \cdot \left(c\_p \cdot -0.5 + c\_n \cdot 0.5\right)\right)}} \]

8. Taylor expanded in s around 0

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right) + t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)}\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right) + s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_n + \frac{-1}{2} \cdot c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_n + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   5. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_n - \frac{1}{2} \cdot c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   6. distribute-lft-out-- N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot \left(c\_n - c\_p\right)\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\left(t \cdot \frac{1}{2}\right) \cdot \left(c\_n - c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\left(\frac{1}{2} \cdot t\right) \cdot \left(c\_n - c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot t\right), \left(c\_n - c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \frac{1}{2}\right), \left(c\_n - c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \left(c\_n - c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(\left(-1 \cdot \frac{1}{2}\right) \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(-1 \cdot \left(\frac{1}{2} \cdot c\_n\right) + \frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(-1 \cdot \left(\frac{1}{2} \cdot c\_n\right) + \left(-1 \cdot \frac{-1}{2}\right) \cdot c\_p\right)\right)\right)\right) \]

   17. associate-*r* N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(-1 \cdot \left(\frac{1}{2} \cdot c\_n\right) + -1 \cdot \left(\frac{-1}{2} \cdot c\_p\right)\right)\right)\right)\right) \]

   18. distribute-lft-in N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(-1 \cdot \left(\frac{1}{2} \cdot c\_n + \frac{-1}{2} \cdot c\_p\right)\right)\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(-1 \cdot \left(\frac{1}{2} \cdot c\_n + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c\_p\right)\right)\right)\right)\right) \]

   20. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(-1 \cdot \left(\frac{1}{2} \cdot c\_n - \frac{1}{2} \cdot c\_p\right)\right)\right)\right)\right) \]

   21. distribute-lft-out-- N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(-1 \cdot \left(\frac{1}{2} \cdot \left(c\_n - c\_p\right)\right)\right)\right)\right)\right) \]

   22. associate-*r* N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(\left(-1 \cdot \frac{1}{2}\right) \cdot \left(c\_n - c\_p\right)\right)\right)\right)\right) \]

   23. metadata-eval N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right), \mathsf{*.f64}\left(s, \left(\frac{-1}{2} \cdot \left(c\_n - c\_p\right)\right)\right)\right)\right) \]

10. Simplified 100.0%

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

11. Taylor expanded in t around 0

    \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \left(s \cdot \left(c\_n - c\_p\right)\right)\right)}\right) \]
12. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(s \cdot \left(c\_n - c\_p\right)\right)\right)\right) \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(s, \left(c\_n - c\_p\right)\right)\right)\right) \]

    3. --lowering--.f64 98.8%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(s, \mathsf{\_.f64}\left(c\_n, c\_p\right)\right)\right)\right) \]

13. Simplified 98.8%

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

14. Final simplification 98.8%

    \[\leadsto e^{-0.5 \cdot \left(\left(c\_n - c\_p\right) \cdot s\right)} \]
15. Add Preprocessing

Alternative 3: 95.7% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(t \cdot 0.5\right) \cdot c\_n} \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 (exp (* (* t 0.5) c_n)))

C

double code(double c_p, double c_n, double t, double s) {
	return exp(((t * 0.5) * 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 = exp(((t * 0.5d0) * c_n))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp(((t * 0.5) * c_n));
}

Python

def code(c_p, c_n, t, s):
	return math.exp(((t * 0.5) * c_n))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(Float64(t * 0.5) * c_n))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = exp(((t * 0.5) * c_n));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(N[(t * 0.5), $MachinePrecision] * c$95$n), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 91.1%

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

4. Applied egg-rr 96.8%

   \[\leadsto \color{blue}{e^{c\_p \cdot \left(\left(0 - \mathsf{log1p}\left(\frac{1}{e^{s}}\right)\right) - \left(0 - \mathsf{log1p}\left(e^{0 - t}\right)\right)\right) + c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{s}}}\right) - \mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{t}}}\right)\right)}} \]

5. Taylor expanded in t around 0

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(c\_n \cdot \left(\log \left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right) - \log \frac{1}{2}\right) + \left(c\_p \cdot \left(\log 2 - \log \left(1 + \frac{1}{e^{s}}\right)\right) + t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(c\_n \cdot \left(\log \left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right) - \log \frac{1}{2}\right)\right), \left(c\_p \cdot \left(\log 2 - \log \left(1 + \frac{1}{e^{s}}\right)\right) + t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)\right)\right) \]

7. Simplified 99.6%

   \[\leadsto e^{\color{blue}{c\_n \cdot \left(\mathsf{log1p}\left(\frac{-1}{1 + e^{0 - s}}\right) - \log 0.5\right) + \left(c\_p \cdot \left(\log 2 - \mathsf{log1p}\left(e^{0 - s}\right)\right) + t \cdot \left(c\_p \cdot -0.5 + c\_n \cdot 0.5\right)\right)}} \]

8. Taylor expanded in s around 0

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

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_n + \frac{-1}{2} \cdot c\_p\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_n + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c\_p\right)\right)\right) \]

   4. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_n - \frac{1}{2} \cdot c\_p\right)\right)\right) \]

   5. distribute-lft-out-- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot \left(c\_n - c\_p\right)\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(t \cdot \frac{1}{2}\right) \cdot \left(c\_n - c\_p\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(\frac{1}{2} \cdot t\right) \cdot \left(c\_n - c\_p\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot t\right), \left(c\_n - c\_p\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \frac{1}{2}\right), \left(c\_n - c\_p\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \left(c\_n - c\_p\right)\right)\right) \]

   11. --lowering--.f64 95.0%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right)\right) \]

10. Simplified 95.0%

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

11. Taylor expanded in c_n around inf

    \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \color{blue}{c\_n}\right)\right) \]
12. Step-by-step derivation

13. Simplified 94.5%

    \[\leadsto e^{\left(t \cdot 0.5\right) \cdot \color{blue}{c\_n}} \]
14. Add Preprocessing

Alternative 4: 94.2% accurate, 39.8× speedup

Math

\[\begin{array}{l} \\ 1 + t \cdot \left(t \cdot \left(\left(c\_n - c\_p\right) \cdot \left(\left(c\_n - c\_p\right) \cdot 0.125\right)\right) + 0.5 \cdot \left(c\_n - c\_p\right)\right) \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (+
  1.0
  (* t (+ (* t (* (- c_n c_p) (* (- c_n c_p) 0.125))) (* 0.5 (- c_n c_p))))))

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (t * ((t * ((c_n - c_p) * ((c_n - c_p) * 0.125))) + (0.5 * (c_n - c_p))));
}

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 + (t * ((t * ((c_n - c_p) * ((c_n - c_p) * 0.125d0))) + (0.5d0 * (c_n - c_p))))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (t * ((t * ((c_n - c_p) * ((c_n - c_p) * 0.125))) + (0.5 * (c_n - c_p))));
}

Python

def code(c_p, c_n, t, s):
	return 1.0 + (t * ((t * ((c_n - c_p) * ((c_n - c_p) * 0.125))) + (0.5 * (c_n - c_p))))

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 + Float64(t * Float64(Float64(t * Float64(Float64(c_n - c_p) * Float64(Float64(c_n - c_p) * 0.125))) + Float64(0.5 * Float64(c_n - c_p)))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 + (t * ((t * ((c_n - c_p) * ((c_n - c_p) * 0.125))) + (0.5 * (c_n - c_p))));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(1.0 + N[(t * N[(N[(t * N[(N[(c$95$n - c$95$p), $MachinePrecision] * N[(N[(c$95$n - c$95$p), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c$95$n - c$95$p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

4. Applied egg-rr 96.8%

   \[\leadsto \color{blue}{e^{c\_p \cdot \left(\left(0 - \mathsf{log1p}\left(\frac{1}{e^{s}}\right)\right) - \left(0 - \mathsf{log1p}\left(e^{0 - t}\right)\right)\right) + c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{s}}}\right) - \mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{t}}}\right)\right)}} \]

5. Taylor expanded in t around 0

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(c\_n \cdot \left(\log \left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right) - \log \frac{1}{2}\right) + \left(c\_p \cdot \left(\log 2 - \log \left(1 + \frac{1}{e^{s}}\right)\right) + t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(c\_n \cdot \left(\log \left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right) - \log \frac{1}{2}\right)\right), \left(c\_p \cdot \left(\log 2 - \log \left(1 + \frac{1}{e^{s}}\right)\right) + t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)\right)\right) \]

7. Simplified 99.6%

   \[\leadsto e^{\color{blue}{c\_n \cdot \left(\mathsf{log1p}\left(\frac{-1}{1 + e^{0 - s}}\right) - \log 0.5\right) + \left(c\_p \cdot \left(\log 2 - \mathsf{log1p}\left(e^{0 - s}\right)\right) + t \cdot \left(c\_p \cdot -0.5 + c\_n \cdot 0.5\right)\right)}} \]

8. Taylor expanded in s around 0

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

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_n + \frac{-1}{2} \cdot c\_p\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_n + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c\_p\right)\right)\right) \]

   4. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_n - \frac{1}{2} \cdot c\_p\right)\right)\right) \]

   5. distribute-lft-out-- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot \left(c\_n - c\_p\right)\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(t \cdot \frac{1}{2}\right) \cdot \left(c\_n - c\_p\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(\frac{1}{2} \cdot t\right) \cdot \left(c\_n - c\_p\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot t\right), \left(c\_n - c\_p\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \frac{1}{2}\right), \left(c\_n - c\_p\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \left(c\_n - c\_p\right)\right)\right) \]

   11. --lowering--.f64 95.0%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right)\right) \]

10. Simplified 95.0%

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

11. Taylor expanded in t around 0

    \[\leadsto \color{blue}{1 + t \cdot \left(\frac{1}{8} \cdot \left(t \cdot {\left(c\_n - c\_p\right)}^{2}\right) + \frac{1}{2} \cdot \left(c\_n - c\_p\right)\right)} \]
12. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(t \cdot \left(\frac{1}{8} \cdot \left(t \cdot {\left(c\_n - c\_p\right)}^{2}\right) + \frac{1}{2} \cdot \left(c\_n - c\_p\right)\right)\right)}\right) \]

    2. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(t \cdot \left(\frac{1}{2} \cdot \left(c\_n - c\_p\right) + \color{blue}{\frac{1}{8} \cdot \left(t \cdot {\left(c\_n - c\_p\right)}^{2}\right)}\right)\right)\right) \]

    3. distribute-lft-out-- N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(t \cdot \left(\left(\frac{1}{2} \cdot c\_n - \frac{1}{2} \cdot c\_p\right) + \color{blue}{\frac{1}{8}} \cdot \left(t \cdot {\left(c\_n - c\_p\right)}^{2}\right)\right)\right)\right) \]

    4. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(t \cdot \left(\left(\frac{1}{2} \cdot c\_n + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c\_p\right) + \color{blue}{\frac{1}{8}} \cdot \left(t \cdot {\left(c\_n - c\_p\right)}^{2}\right)\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(t \cdot \left(\left(\frac{1}{2} \cdot c\_n + \frac{-1}{2} \cdot c\_p\right) + \frac{1}{8} \cdot \left(t \cdot {\left(c\_n - c\_p\right)}^{2}\right)\right)\right)\right) \]

    6. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(t \cdot \left(\left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right) + \color{blue}{\frac{1}{8}} \cdot \left(t \cdot {\left(c\_n - c\_p\right)}^{2}\right)\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \color{blue}{\left(\left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right) + \frac{1}{8} \cdot \left(t \cdot {\left(c\_n - c\_p\right)}^{2}\right)\right)}\right)\right) \]

    8. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \left(\frac{1}{8} \cdot \left(t \cdot {\left(c\_n - c\_p\right)}^{2}\right) + \color{blue}{\left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)}\right)\right)\right) \]

    9. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{1}{8} \cdot \left(t \cdot {\left(c\_n - c\_p\right)}^{2}\right)\right), \color{blue}{\left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)}\right)\right)\right) \]

13. Simplified 93.9%

    \[\leadsto \color{blue}{1 + t \cdot \left(t \cdot \left(\left(c\_n - c\_p\right) \cdot \left(\left(c\_n - c\_p\right) \cdot 0.125\right)\right) + \left(c\_n - c\_p\right) \cdot 0.5\right)} \]

14. Final simplification 93.9%

    \[\leadsto 1 + t \cdot \left(t \cdot \left(\left(c\_n - c\_p\right) \cdot \left(\left(c\_n - c\_p\right) \cdot 0.125\right)\right) + 0.5 \cdot \left(c\_n - c\_p\right)\right) \]
15. Add Preprocessing

Alternative 5: 94.2% accurate, 92.8× speedup

Math

\[\begin{array}{l} \\ \left(t \cdot 0.5\right) \cdot \left(c\_n - c\_p\right) + 1 \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 (+ (* (* t 0.5) (- c_n c_p)) 1.0))

C

double code(double c_p, double c_n, double t, double s) {
	return ((t * 0.5) * (c_n - c_p)) + 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 = ((t * 0.5d0) * (c_n - c_p)) + 1.0d0
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return ((t * 0.5) * (c_n - c_p)) + 1.0;
}

Python

def code(c_p, c_n, t, s):
	return ((t * 0.5) * (c_n - c_p)) + 1.0

Julia

function code(c_p, c_n, t, s)
	return Float64(Float64(Float64(t * 0.5) * Float64(c_n - c_p)) + 1.0)
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = ((t * 0.5) * (c_n - c_p)) + 1.0;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(N[(N[(t * 0.5), $MachinePrecision] * N[(c$95$n - c$95$p), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

4. Applied egg-rr 96.8%

   \[\leadsto \color{blue}{e^{c\_p \cdot \left(\left(0 - \mathsf{log1p}\left(\frac{1}{e^{s}}\right)\right) - \left(0 - \mathsf{log1p}\left(e^{0 - t}\right)\right)\right) + c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{s}}}\right) - \mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{t}}}\right)\right)}} \]

5. Taylor expanded in t around 0

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(c\_n \cdot \left(\log \left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right) - \log \frac{1}{2}\right) + \left(c\_p \cdot \left(\log 2 - \log \left(1 + \frac{1}{e^{s}}\right)\right) + t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(c\_n \cdot \left(\log \left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right) - \log \frac{1}{2}\right)\right), \left(c\_p \cdot \left(\log 2 - \log \left(1 + \frac{1}{e^{s}}\right)\right) + t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)\right)\right) \]

7. Simplified 99.6%

   \[\leadsto e^{\color{blue}{c\_n \cdot \left(\mathsf{log1p}\left(\frac{-1}{1 + e^{0 - s}}\right) - \log 0.5\right) + \left(c\_p \cdot \left(\log 2 - \mathsf{log1p}\left(e^{0 - s}\right)\right) + t \cdot \left(c\_p \cdot -0.5 + c\_n \cdot 0.5\right)\right)}} \]

8. Taylor expanded in s around 0

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

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_n + \frac{-1}{2} \cdot c\_p\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_n + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c\_p\right)\right)\right) \]

   4. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_n - \frac{1}{2} \cdot c\_p\right)\right)\right) \]

   5. distribute-lft-out-- N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot \left(c\_n - c\_p\right)\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(t \cdot \frac{1}{2}\right) \cdot \left(c\_n - c\_p\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\left(\frac{1}{2} \cdot t\right) \cdot \left(c\_n - c\_p\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot t\right), \left(c\_n - c\_p\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(t \cdot \frac{1}{2}\right), \left(c\_n - c\_p\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \left(c\_n - c\_p\right)\right)\right) \]

   11. --lowering--.f64 95.0%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(t, \frac{1}{2}\right), \mathsf{\_.f64}\left(c\_n, c\_p\right)\right)\right) \]

10. Simplified 95.0%

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

11. Taylor expanded in t around 0

    \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot \left(t \cdot \left(c\_n - c\_p\right)\right)} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto 1 + \left(t \cdot \left(c\_n - c\_p\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

    2. associate-*r* N/A

       \[\leadsto 1 + t \cdot \color{blue}{\left(\left(c\_n - c\_p\right) \cdot \frac{1}{2}\right)} \]

    3. *-commutative N/A

       \[\leadsto 1 + t \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(c\_n - c\_p\right)}\right) \]

    4. distribute-lft-out-- N/A

       \[\leadsto 1 + t \cdot \left(\frac{1}{2} \cdot c\_n - \color{blue}{\frac{1}{2} \cdot c\_p}\right) \]

    5. cancel-sign-sub-inv N/A

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

    6. metadata-eval N/A

       \[\leadsto 1 + t \cdot \left(\frac{1}{2} \cdot c\_n + \frac{-1}{2} \cdot c\_p\right) \]

    7. +-commutative N/A

       \[\leadsto 1 + t \cdot \left(\frac{-1}{2} \cdot c\_p + \color{blue}{\frac{1}{2} \cdot c\_n}\right) \]

    8. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(t \cdot \left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right)\right)}\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{2} \cdot c\_p + \frac{1}{2} \cdot c\_n\right) \cdot \color{blue}{t}\right)\right) \]

    10. +-commutative N/A

        \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot c\_n + \frac{-1}{2} \cdot c\_p\right) \cdot t\right)\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot c\_n + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c\_p\right) \cdot t\right)\right) \]

    12. cancel-sign-sub-inv N/A

        \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot c\_n - \frac{1}{2} \cdot c\_p\right) \cdot t\right)\right) \]

    13. distribute-lft-out-- N/A

        \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot \left(c\_n - c\_p\right)\right) \cdot t\right)\right) \]

    14. *-commutative N/A

        \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\left(c\_n - c\_p\right) \cdot \frac{1}{2}\right) \cdot t\right)\right) \]

    15. associate-*l* N/A

        \[\leadsto \mathsf{+.f64}\left(1, \left(\left(c\_n - c\_p\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot t\right)}\right)\right) \]

    16. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(c\_n - c\_p\right), \color{blue}{\left(\frac{1}{2} \cdot t\right)}\right)\right) \]

    17. --lowering--.f64 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(c\_n, c\_p\right), \left(\color{blue}{\frac{1}{2}} \cdot t\right)\right)\right) \]

    18. *-lowering-*.f64 93.8%

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(c\_n, c\_p\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{t}\right)\right)\right) \]

13. Simplified 93.8%

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

14. Final simplification 93.8%

    \[\leadsto \left(t \cdot 0.5\right) \cdot \left(c\_n - c\_p\right) + 1 \]
15. Add Preprocessing

Alternative 6: 94.2% accurate, 119.3× speedup

Math

\[\begin{array}{l} \\ 1 + -0.5 \cdot \left(t \cdot c\_p\right) \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 (+ 1.0 (* -0.5 (* t c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (-0.5 * (t * c_p));
}

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 + ((-0.5d0) * (t * c_p))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (-0.5 * (t * c_p));
}

Python

def code(c_p, c_n, t, s):
	return 1.0 + (-0.5 * (t * c_p))

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 + Float64(-0.5 * Float64(t * c_p)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 + (-0.5 * (t * c_p));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(1.0 + N[(-0.5 * N[(t * c$95$p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in c_n around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}\right), \color{blue}{\left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)}\right) \]

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right), c\_p\right), \left({\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_p}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right), c\_p\right), \left({\left(\frac{\color{blue}{1}}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

   5. exp-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{s}\right)\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

   7. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

   8. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right), \color{blue}{c\_p}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\mathsf{neg}\left(t\right)}\right)\right), c\_p\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(t\right)}\right)\right)\right), c\_p\right)\right) \]

   11. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), c\_p\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(0 - t\right)\right)\right)\right), c\_p\right)\right) \]

   13. --lowering--.f64 92.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, t\right)\right)\right)\right), c\_p\right)\right) \]

5. Simplified 92.8%

   \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + \frac{1}{e^{s}}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{0 - t}}\right)}^{c\_p}}} \]

6. Taylor expanded in s around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\frac{1}{2}}^{c\_p}\right), \color{blue}{\left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)}\right) \]

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\frac{1}{2}, c\_p\right), \left({\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_p}\right)\right) \]

   3. neg-mul-1 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\frac{1}{2}, c\_p\right), \left({\left(\frac{1}{1 + e^{-1 \cdot t}}\right)}^{c\_p}\right)\right) \]

   4. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\frac{1}{2}, c\_p\right), \mathsf{pow.f64}\left(\left(\frac{1}{1 + e^{-1 \cdot t}}\right), \color{blue}{c\_p}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\frac{1}{2}, c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{-1 \cdot t}\right)\right), c\_p\right)\right) \]

   6. neg-mul-1 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\frac{1}{2}, c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\mathsf{neg}\left(t\right)}\right)\right), c\_p\right)\right) \]

   7. rec-exp N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\frac{1}{2}, c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \frac{1}{e^{t}}\right)\right), c\_p\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\frac{1}{2}, c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{t}}\right)\right)\right), c\_p\right)\right) \]

   9. rec-exp N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\frac{1}{2}, c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(t\right)}\right)\right)\right), c\_p\right)\right) \]

   10. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\frac{1}{2}, c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), c\_p\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\frac{1}{2}, c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(0 - t\right)\right)\right)\right), c\_p\right)\right) \]

   12. --lowering--.f64 91.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\frac{1}{2}, c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, t\right)\right)\right)\right), c\_p\right)\right) \]

8. Simplified 91.3%

   \[\leadsto \color{blue}{\frac{{0.5}^{c\_p}}{{\left(\frac{1}{1 + e^{0 - t}}\right)}^{c\_p}}} \]

9. Taylor expanded in t around 0

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

    1. +-commutative N/A

       \[\leadsto \frac{-1}{2} \cdot \left(c\_p \cdot t\right) + \color{blue}{1} \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(c\_p \cdot t\right)\right), \color{blue}{1}\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(c\_p \cdot t\right)\right), 1\right) \]

    4. *-lowering-*.f64 93.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(c\_p, t\right)\right), 1\right) \]

11. Simplified 93.7%

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

12. Final simplification 93.7%

    \[\leadsto 1 + -0.5 \cdot \left(t \cdot c\_p\right) \]
13. Add Preprocessing

Alternative 7: 94.2% accurate, 835.0× 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 91.1%

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

3. Taylor expanded in c_n around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}\right), \color{blue}{\left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)}\right) \]

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right), c\_p\right), \left({\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_p}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right), c\_p\right), \left({\left(\frac{\color{blue}{1}}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

   5. exp-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{s}\right)\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

   7. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

   8. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right), \color{blue}{c\_p}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\mathsf{neg}\left(t\right)}\right)\right), c\_p\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(t\right)}\right)\right)\right), c\_p\right)\right) \]

   11. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), c\_p\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(0 - t\right)\right)\right)\right), c\_p\right)\right) \]

   13. --lowering--.f64 92.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, t\right)\right)\right)\right), c\_p\right)\right) \]

5. Simplified 92.8%

   \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + \frac{1}{e^{s}}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{0 - t}}\right)}^{c\_p}}} \]

6. Taylor expanded in c_p around 0

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

8. Simplified 93.5%

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

Developer Target 1: 96.4% accurate, 1.3× 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 2024184 
(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))))