Harley's example

Percentage Accurate: 91.3% → 97.8%

Time: 1.1min

Alternatives: 6

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 5.009801579707393e-308
  2. c_n = 3.049334375824566e-104
  3. t = 2.6113079547077797e-162
  4. s = 2.3938849183613884e+291

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_p, c_n, t, s)
use fmin_fmax_functions
    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: 91.3% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_p, c_n, t, s)
use fmin_fmax_functions
    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: 97.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp
  (-
   (* (- (* s (+ 0.5 (* -0.125 s))) (log 2.0)) c_p)
   (* (- (* (fma -0.125 t 0.5) t) (log 2.0)) c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp(((((s * (0.5 + (-0.125 * s))) - log(2.0)) * c_p) - (((fma(-0.125, t, 0.5) * t) - log(2.0)) * c_p)));
}

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(Float64(Float64(Float64(s * Float64(0.5 + Float64(-0.125 * s))) - log(2.0)) * c_p) - Float64(Float64(Float64(fma(-0.125, t, 0.5) * t) - log(2.0)) * c_p)))
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(N[(N[(N[(s * N[(0.5 + N[(-0.125 * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision] * c$95$p), $MachinePrecision] - N[(N[(N[(N[(-0.125 * t + 0.5), $MachinePrecision] * t), $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision] * c$95$p), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 92.3%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
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. pow-to-exp N/A

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

   2. *-commutative N/A

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

   3. pow-to-exp N/A

      \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot c\_p}} \]

   4. *-commutative N/A

      \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}} \]

   5. div-exp N/A

      \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

   6. lower-exp.f64 N/A

      \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

   7. lower--.f64 N/A

      \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

5. Applied rewrites 95.3%

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

6. Taylor expanded in s around 0

   \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]
7. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

   2. lower-*.f64 N/A

      \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

   3. lower-+.f64 N/A

      \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

   4. lower-*.f64 N/A

      \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

   5. lower-log.f64 97.3

      \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

8. Applied rewrites 97.3%

   \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

9. Taylor expanded in t around 0

   \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(t \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) - \log 2\right) \cdot c\_p} \]
10. Step-by-step derivation

    1. lower--.f64 N/A

       \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(t \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) - \log 2\right) \cdot c\_p} \]

    2. *-commutative N/A

       \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) \cdot t - \log 2\right) \cdot c\_p} \]

    3. lower-*.f64 N/A

       \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) \cdot t - \log 2\right) \cdot c\_p} \]

    4. +-commutative N/A

       \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\left(\frac{-1}{8} \cdot t + \frac{1}{2}\right) \cdot t - \log 2\right) \cdot c\_p} \]

    5. lower-fma.f64 N/A

       \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(\frac{-1}{8}, t, \frac{1}{2}\right) \cdot t - \log 2\right) \cdot c\_p} \]

    6. lift-log.f64 98.1

       \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p} \]

11. Applied rewrites 98.1%

    \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p} \]
12. Add Preprocessing

Alternative 2: 96.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := \frac{1}{1 + e^{-s}}\\ \mathbf{if}\;\frac{{t\_2}^{c\_p} \cdot {\left(1 - t\_2\right)}^{c\_n}}{{t\_1}^{c\_p} \cdot {\left(1 - t\_1\right)}^{c\_n}} \leq 0:\\ \;\;\;\;e^{\left(-0.125 \cdot \left(s \cdot s\right)\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-c\_p\right) \cdot \left(\mathsf{log1p}\left(1\right) + \left(0.5 \cdot t - \log 2\right)\right)}\\ \end{array} \end{array} \]

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))))))
   (if (<=
        (/
         (* (pow t_2 c_p) (pow (- 1.0 t_2) c_n))
         (* (pow t_1 c_p) (pow (- 1.0 t_1) c_n)))
        0.0)
     (exp
      (-
       (* (* -0.125 (* s s)) c_p)
       (* (- (* (fma -0.125 t 0.5) t) (log 2.0)) c_p)))
     (exp (* (- c_p) (+ (log1p 1.0) (- (* 0.5 t) (log 2.0))))))))

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));
	double tmp;
	if (((pow(t_2, c_p) * pow((1.0 - t_2), c_n)) / (pow(t_1, c_p) * pow((1.0 - t_1), c_n))) <= 0.0) {
		tmp = exp((((-0.125 * (s * s)) * c_p) - (((fma(-0.125, t, 0.5) * t) - log(2.0)) * c_p)));
	} else {
		tmp = exp((-c_p * (log1p(1.0) + ((0.5 * t) - log(2.0)))));
	}
	return tmp;
}

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))))
	tmp = 0.0
	if (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))) <= 0.0)
		tmp = exp(Float64(Float64(Float64(-0.125 * Float64(s * s)) * c_p) - Float64(Float64(Float64(fma(-0.125, t, 0.5) * t) - log(2.0)) * c_p)));
	else
		tmp = exp(Float64(Float64(-c_p) * Float64(log1p(1.0) + Float64(Float64(0.5 * t) - log(2.0)))));
	end
	return tmp
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]}, If[LessEqual[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], 0.0], N[Exp[N[(N[(N[(-0.125 * N[(s * s), $MachinePrecision]), $MachinePrecision] * c$95$p), $MachinePrecision] - N[(N[(N[(N[(-0.125 * t + 0.5), $MachinePrecision] * t), $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision] * c$95$p), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[((-c$95$p) * N[(N[Log[1 + 1.0], $MachinePrecision] + N[(N[(0.5 * t), $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n))) < 0.0

   1. Initial program 56.0%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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. pow-to-exp N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

         \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot c\_p}} \]

      4. *-commutative N/A

         \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}} \]

      5. div-exp N/A

         \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

      7. lower--.f64 N/A

         \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in s around 0

      \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

      2. lower-*.f64 N/A

         \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

      3. lower-+.f64 N/A

         \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

      4. lower-*.f64 N/A

         \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

      5. lower-log.f64 95.1

         \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

   8. Applied rewrites 95.1%

      \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

   9. Taylor expanded in t around 0

      \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(t \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) - \log 2\right) \cdot c\_p} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(t \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) - \log 2\right) \cdot c\_p} \]

       2. *-commutative N/A

          \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) \cdot t - \log 2\right) \cdot c\_p} \]

       3. lower-*.f64 N/A

          \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) \cdot t - \log 2\right) \cdot c\_p} \]

       4. +-commutative N/A

          \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\left(\frac{-1}{8} \cdot t + \frac{1}{2}\right) \cdot t - \log 2\right) \cdot c\_p} \]

       5. lower-fma.f64 N/A

          \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(\frac{-1}{8}, t, \frac{1}{2}\right) \cdot t - \log 2\right) \cdot c\_p} \]

       6. lift-log.f64 95.1

          \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p} \]

   11. Applied rewrites 95.1%

       \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p} \]

   12. Taylor expanded in s around inf

       \[\leadsto e^{\left(\frac{-1}{8} \cdot {s}^{2}\right) \cdot c\_p - \left(\mathsf{fma}\left(\frac{-1}{8}, t, \frac{1}{2}\right) \cdot t - \log 2\right) \cdot c\_p} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto e^{\left(\frac{-1}{8} \cdot {s}^{2}\right) \cdot c\_p - \left(\mathsf{fma}\left(\frac{-1}{8}, t, \frac{1}{2}\right) \cdot t - \log 2\right) \cdot c\_p} \]

       2. unpow2 N/A

          \[\leadsto e^{\left(\frac{-1}{8} \cdot \left(s \cdot s\right)\right) \cdot c\_p - \left(\mathsf{fma}\left(\frac{-1}{8}, t, \frac{1}{2}\right) \cdot t - \log 2\right) \cdot c\_p} \]

       3. lower-*.f64 95.1

          \[\leadsto e^{\left(-0.125 \cdot \left(s \cdot s\right)\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p} \]

   14. Applied rewrites 95.1%

       \[\leadsto e^{\left(-0.125 \cdot \left(s \cdot s\right)\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p} \]

   if 0.0 < (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n)))

   1. Initial program 93.9%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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. pow-to-exp N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

         \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot c\_p}} \]

      4. *-commutative N/A

         \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}} \]

      5. div-exp N/A

         \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

      7. lower--.f64 N/A

         \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in t around 0

      \[\leadsto e^{\left(-\mathsf{log1p}\left(e^{-s}\right)\right) \cdot c\_p - \left(\frac{1}{2} \cdot t - \log 2\right) \cdot c\_p} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-log.f64 98.2

         \[\leadsto e^{\left(-\mathsf{log1p}\left(e^{-s}\right)\right) \cdot c\_p - \left(0.5 \cdot t - \log 2\right) \cdot c\_p} \]

   8. Applied rewrites 98.2%

      \[\leadsto e^{\left(-\mathsf{log1p}\left(e^{-s}\right)\right) \cdot c\_p - \left(0.5 \cdot t - \log 2\right) \cdot c\_p} \]

   9. Taylor expanded in s around 0

      \[\leadsto e^{\left(-\mathsf{log1p}\left(1\right)\right) \cdot c\_p - \left(\frac{1}{2} \cdot t - \log 2\right) \cdot c\_p} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.2%

       \[\leadsto e^{\left(-\mathsf{log1p}\left(1\right)\right) \cdot c\_p - \left(0.5 \cdot t - \log 2\right) \cdot c\_p} \]
   12. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto e^{\left(-\mathsf{log1p}\left(1\right)\right) \cdot c\_p - \left(\frac{1}{2} \cdot t - \log 2\right) \cdot c\_p} \]

       2. lift-*.f64 N/A

          \[\leadsto e^{\left(-\mathsf{log1p}\left(1\right)\right) \cdot c\_p - \left(\frac{1}{2} \cdot t - \log 2\right) \cdot c\_p} \]

       3. lift-*.f64 N/A

          \[\leadsto e^{\left(-\mathsf{log1p}\left(1\right)\right) \cdot c\_p - \left(\frac{1}{2} \cdot t - \log 2\right) \cdot c\_p} \]

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

          \[\leadsto e^{c\_p \cdot \left(\left(-\mathsf{log1p}\left(1\right)\right) - \left(\frac{1}{2} \cdot t - \log 2\right)\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto e^{c\_p \cdot \left(\left(-\mathsf{log1p}\left(1\right)\right) - \left(\frac{1}{2} \cdot t - \log 2\right)\right)} \]

   13. Applied rewrites 98.2%

       \[\leadsto \color{blue}{e^{c\_p \cdot \left(\left(-\mathsf{log1p}\left(1\right)\right) - \left(0.5 \cdot t - \log 2\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \leq 0:\\ \;\;\;\;e^{\left(-0.125 \cdot \left(s \cdot s\right)\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-c\_p\right) \cdot \left(\mathsf{log1p}\left(1\right) + \left(0.5 \cdot t - \log 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := \frac{1}{1 + e^{-s}}\\ \mathbf{if}\;\frac{{t\_2}^{c\_p} \cdot {\left(1 - t\_2\right)}^{c\_n}}{{t\_1}^{c\_p} \cdot {\left(1 - t\_1\right)}^{c\_n}} \leq 0:\\ \;\;\;\;e^{\left(-0.125 \cdot \left(s \cdot s\right)\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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))))))
   (if (<=
        (/
         (* (pow t_2 c_p) (pow (- 1.0 t_2) c_n))
         (* (pow t_1 c_p) (pow (- 1.0 t_1) c_n)))
        0.0)
     (exp
      (-
       (* (* -0.125 (* s s)) c_p)
       (* (- (* (fma -0.125 t 0.5) t) (log 2.0)) c_p)))
     1.0)))

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));
	double tmp;
	if (((pow(t_2, c_p) * pow((1.0 - t_2), c_n)) / (pow(t_1, c_p) * pow((1.0 - t_1), c_n))) <= 0.0) {
		tmp = exp((((-0.125 * (s * s)) * c_p) - (((fma(-0.125, t, 0.5) * t) - log(2.0)) * c_p)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

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))))
	tmp = 0.0
	if (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))) <= 0.0)
		tmp = exp(Float64(Float64(Float64(-0.125 * Float64(s * s)) * c_p) - Float64(Float64(Float64(fma(-0.125, t, 0.5) * t) - log(2.0)) * c_p)));
	else
		tmp = 1.0;
	end
	return tmp
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]}, If[LessEqual[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], 0.0], N[Exp[N[(N[(N[(-0.125 * N[(s * s), $MachinePrecision]), $MachinePrecision] * c$95$p), $MachinePrecision] - N[(N[(N[(N[(-0.125 * t + 0.5), $MachinePrecision] * t), $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision] * c$95$p), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n))) < 0.0

   1. Initial program 56.0%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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. pow-to-exp N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

         \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot c\_p}} \]

      4. *-commutative N/A

         \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}} \]

      5. div-exp N/A

         \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

      7. lower--.f64 N/A

         \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in s around 0

      \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

      2. lower-*.f64 N/A

         \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

      3. lower-+.f64 N/A

         \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

      4. lower-*.f64 N/A

         \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

      5. lower-log.f64 95.1

         \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

   8. Applied rewrites 95.1%

      \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

   9. Taylor expanded in t around 0

      \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(t \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) - \log 2\right) \cdot c\_p} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(t \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) - \log 2\right) \cdot c\_p} \]

       2. *-commutative N/A

          \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) \cdot t - \log 2\right) \cdot c\_p} \]

       3. lower-*.f64 N/A

          \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) \cdot t - \log 2\right) \cdot c\_p} \]

       4. +-commutative N/A

          \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\left(\frac{-1}{8} \cdot t + \frac{1}{2}\right) \cdot t - \log 2\right) \cdot c\_p} \]

       5. lower-fma.f64 N/A

          \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(\frac{-1}{8}, t, \frac{1}{2}\right) \cdot t - \log 2\right) \cdot c\_p} \]

       6. lift-log.f64 95.1

          \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p} \]

   11. Applied rewrites 95.1%

       \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p} \]

   12. Taylor expanded in s around inf

       \[\leadsto e^{\left(\frac{-1}{8} \cdot {s}^{2}\right) \cdot c\_p - \left(\mathsf{fma}\left(\frac{-1}{8}, t, \frac{1}{2}\right) \cdot t - \log 2\right) \cdot c\_p} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto e^{\left(\frac{-1}{8} \cdot {s}^{2}\right) \cdot c\_p - \left(\mathsf{fma}\left(\frac{-1}{8}, t, \frac{1}{2}\right) \cdot t - \log 2\right) \cdot c\_p} \]

       2. unpow2 N/A

          \[\leadsto e^{\left(\frac{-1}{8} \cdot \left(s \cdot s\right)\right) \cdot c\_p - \left(\mathsf{fma}\left(\frac{-1}{8}, t, \frac{1}{2}\right) \cdot t - \log 2\right) \cdot c\_p} \]

       3. lower-*.f64 95.1

          \[\leadsto e^{\left(-0.125 \cdot \left(s \cdot s\right)\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p} \]

   14. Applied rewrites 95.1%

       \[\leadsto e^{\left(-0.125 \cdot \left(s \cdot s\right)\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p} \]

   if 0.0 < (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n)))

   1. Initial program 93.9%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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. pow-to-exp N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

         \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot c\_p}} \]

      4. *-commutative N/A

         \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}} \]

      5. div-exp N/A

         \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

      7. lower--.f64 N/A

         \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in c_p around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 98.2%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ e^{\left(s \cdot 0.5 - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp
  (-
   (* (- (* s 0.5) (log 2.0)) c_p)
   (* (- (* (fma -0.125 t 0.5) t) (log 2.0)) c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp(((((s * 0.5) - log(2.0)) * c_p) - (((fma(-0.125, t, 0.5) * t) - log(2.0)) * c_p)));
}

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(Float64(Float64(Float64(s * 0.5) - log(2.0)) * c_p) - Float64(Float64(Float64(fma(-0.125, t, 0.5) * t) - log(2.0)) * c_p)))
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(N[(N[(N[(s * 0.5), $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision] * c$95$p), $MachinePrecision] - N[(N[(N[(N[(-0.125 * t + 0.5), $MachinePrecision] * t), $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision] * c$95$p), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 92.3%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
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. pow-to-exp N/A

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

   2. *-commutative N/A

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

   3. pow-to-exp N/A

      \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot c\_p}} \]

   4. *-commutative N/A

      \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}} \]

   5. div-exp N/A

      \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

   6. lower-exp.f64 N/A

      \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

   7. lower--.f64 N/A

      \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

5. Applied rewrites 95.3%

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

6. Taylor expanded in s around 0

   \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]
7. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

   2. lower-*.f64 N/A

      \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

   3. lower-+.f64 N/A

      \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

   4. lower-*.f64 N/A

      \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

   5. lower-log.f64 97.3

      \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

8. Applied rewrites 97.3%

   \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(-\mathsf{log1p}\left(e^{-t}\right)\right) \cdot c\_p} \]

9. Taylor expanded in t around 0

   \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(t \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) - \log 2\right) \cdot c\_p} \]
10. Step-by-step derivation

    1. lower--.f64 N/A

       \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(t \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) - \log 2\right) \cdot c\_p} \]

    2. *-commutative N/A

       \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) \cdot t - \log 2\right) \cdot c\_p} \]

    3. lower-*.f64 N/A

       \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\left(\frac{1}{2} + \frac{-1}{8} \cdot t\right) \cdot t - \log 2\right) \cdot c\_p} \]

    4. +-commutative N/A

       \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\left(\frac{-1}{8} \cdot t + \frac{1}{2}\right) \cdot t - \log 2\right) \cdot c\_p} \]

    5. lower-fma.f64 N/A

       \[\leadsto e^{\left(s \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot s\right) - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(\frac{-1}{8}, t, \frac{1}{2}\right) \cdot t - \log 2\right) \cdot c\_p} \]

    6. lift-log.f64 98.1

       \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p} \]

11. Applied rewrites 98.1%

    \[\leadsto e^{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p} \]

12. Taylor expanded in s around 0

    \[\leadsto e^{\left(s \cdot \frac{1}{2} - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(\frac{-1}{8}, t, \frac{1}{2}\right) \cdot t - \log 2\right) \cdot c\_p} \]
13. Step-by-step derivation

14. Applied rewrites 96.9%

    \[\leadsto e^{\left(s \cdot 0.5 - \log 2\right) \cdot c\_p - \left(\mathsf{fma}\left(-0.125, t, 0.5\right) \cdot t - \log 2\right) \cdot c\_p} \]
15. Add Preprocessing

Alternative 5: 96.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{fma}\left(-0.5, s, \log 2\right) \cdot \left(-c\_p\right) - \log 0.5 \cdot c\_p} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp (- (* (fma -0.5 s (log 2.0)) (- c_p)) (* (log 0.5) c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp(((fma(-0.5, s, log(2.0)) * -c_p) - (log(0.5) * c_p)));
}

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(Float64(fma(-0.5, s, log(2.0)) * Float64(-c_p)) - Float64(log(0.5) * c_p)))
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(N[(N[(-0.5 * s + N[Log[2.0], $MachinePrecision]), $MachinePrecision] * (-c$95$p)), $MachinePrecision] - N[(N[Log[0.5], $MachinePrecision] * c$95$p), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 92.3%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
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. pow-to-exp N/A

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

   2. *-commutative N/A

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

   3. pow-to-exp N/A

      \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot c\_p}} \]

   4. *-commutative N/A

      \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}} \]

   5. div-exp N/A

      \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

   6. lower-exp.f64 N/A

      \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

   7. lower--.f64 N/A

      \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

5. Applied rewrites 95.3%

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

6. Taylor expanded in t around 0

   \[\leadsto e^{\left(-\mathsf{log1p}\left(e^{-s}\right)\right) \cdot c\_p - \left(-1 \cdot \log 2\right) \cdot c\_p} \]
7. Step-by-step derivation

   1. log-pow-rev N/A

      \[\leadsto e^{\left(-\mathsf{log1p}\left(e^{-s}\right)\right) \cdot c\_p - \log \left({2}^{-1}\right) \cdot c\_p} \]

   2. metadata-eval N/A

      \[\leadsto e^{\left(-\mathsf{log1p}\left(e^{-s}\right)\right) \cdot c\_p - \log \frac{1}{2} \cdot c\_p} \]

   3. lower-log.f64 96.0

      \[\leadsto e^{\left(-\mathsf{log1p}\left(e^{-s}\right)\right) \cdot c\_p - \log 0.5 \cdot c\_p} \]

8. Applied rewrites 96.0%

   \[\leadsto e^{\left(-\mathsf{log1p}\left(e^{-s}\right)\right) \cdot c\_p - \log 0.5 \cdot c\_p} \]

9. Taylor expanded in s around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

       \[\leadsto e^{\left(-\mathsf{fma}\left(\frac{-1}{2}, s, \log 2\right)\right) \cdot c\_p - \log \frac{1}{2} \cdot c\_p} \]

    3. lift-log.f64 96.9

       \[\leadsto e^{\left(-\mathsf{fma}\left(-0.5, s, \log 2\right)\right) \cdot c\_p - \log 0.5 \cdot c\_p} \]

11. Applied rewrites 96.9%

    \[\leadsto e^{\left(-\mathsf{fma}\left(-0.5, s, \log 2\right)\right) \cdot c\_p - \log 0.5 \cdot c\_p} \]

12. Final simplification 96.9%

    \[\leadsto e^{\mathsf{fma}\left(-0.5, s, \log 2\right) \cdot \left(-c\_p\right) - \log 0.5 \cdot c\_p} \]
13. Add Preprocessing

Alternative 6: 94.0% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_p, c_n, t, s)
use fmin_fmax_functions
    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 92.3%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
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. pow-to-exp N/A

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

   2. *-commutative N/A

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

   3. pow-to-exp N/A

      \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{\log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right) \cdot c\_p}} \]

   4. *-commutative N/A

      \[\leadsto \frac{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}{e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}} \]

   5. div-exp N/A

      \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

   6. lower-exp.f64 N/A

      \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

   7. lower--.f64 N/A

      \[\leadsto e^{c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - c\_p \cdot \log \left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

5. Applied rewrites 95.3%

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

6. Taylor expanded in c_p around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 95.8%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer Target 1: 96.5% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_p, c_n, t, s)
use fmin_fmax_functions
    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 2025027 
(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))))