Harley's example

Percentage Accurate: 90.7% → 99.5%

Time: 47.8s

Alternatives: 7

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 7.024388495549865e-91
  2. c_n = 9.054815484205631e+76
  3. t = 1.3316392056400983e+32
  4. s = 1.0739693017674478e+149

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.7% 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: 99.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log 0.5 \cdot c\_n\\ e^{\mathsf{fma}\left(\left(c\_p - c\_n\right) \cdot t, -0.5, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, c\_n - c\_p, \left(-0.125 \cdot \mathsf{fma}\left(c\_n, 1, c\_p\right)\right) \cdot s\right), s, t\_1\right)\right) - t\_1} \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (* (log 0.5) c_n)))
   (exp
    (-
     (fma
      (* (- c_p c_n) t)
      -0.5
      (fma (fma -0.5 (- c_n c_p) (* (* -0.125 (fma c_n 1.0 c_p)) s)) s t_1))
     t_1))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = log(0.5) * c_n;
	return exp((fma(((c_p - c_n) * t), -0.5, fma(fma(-0.5, (c_n - c_p), ((-0.125 * fma(c_n, 1.0, c_p)) * s)), s, t_1)) - t_1));
}

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(log(0.5) * c_n)
	return exp(Float64(fma(Float64(Float64(c_p - c_n) * t), -0.5, fma(fma(-0.5, Float64(c_n - c_p), Float64(Float64(-0.125 * fma(c_n, 1.0, c_p)) * s)), s, t_1)) - t_1))
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(N[Log[0.5], $MachinePrecision] * c$95$n), $MachinePrecision]}, N[Exp[N[(N[(N[(N[(c$95$p - c$95$n), $MachinePrecision] * t), $MachinePrecision] * -0.5 + N[(N[(-0.5 * N[(c$95$n - c$95$p), $MachinePrecision] + N[(N[(-0.125 * N[(c$95$n * 1.0 + c$95$p), $MachinePrecision]), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision] * s + t$95$1), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 88.4%

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

4. Applied rewrites 93.7%

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

5. Taylor expanded in t around 0

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

7. Applied rewrites 97.3%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 99.4%

    \[\leadsto e^{\mathsf{fma}\left(\left(c\_p - c\_n\right) \cdot t, -0.5, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, c\_n \cdot 1 - c\_p, \left(-0.125 \cdot \mathsf{fma}\left(c\_n, 1, c\_p\right)\right) \cdot s\right), s, \log 0.5 \cdot c\_n\right)\right) - \color{blue}{\log 0.5 \cdot c\_n}} \]

11. Final simplification 99.4%

    \[\leadsto e^{\mathsf{fma}\left(\left(c\_p - c\_n\right) \cdot t, -0.5, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, c\_n - c\_p, \left(-0.125 \cdot \mathsf{fma}\left(c\_n, 1, c\_p\right)\right) \cdot s\right), s, \log 0.5 \cdot c\_n\right)\right) - \log 0.5 \cdot c\_n} \]
12. Add Preprocessing

Alternative 2: 99.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log 0.5 \cdot c\_n\\ e^{\mathsf{fma}\left(\left(c\_p - c\_n\right) \cdot t, -0.5, \mathsf{fma}\left(-0.5 \cdot \left(c\_n - c\_p\right), s, t\_1\right)\right) - t\_1} \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (* (log 0.5) c_n)))
   (exp
    (- (fma (* (- c_p c_n) t) -0.5 (fma (* -0.5 (- c_n c_p)) s t_1)) t_1))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = log(0.5) * c_n;
	return exp((fma(((c_p - c_n) * t), -0.5, fma((-0.5 * (c_n - c_p)), s, t_1)) - t_1));
}

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(log(0.5) * c_n)
	return exp(Float64(fma(Float64(Float64(c_p - c_n) * t), -0.5, fma(Float64(-0.5 * Float64(c_n - c_p)), s, t_1)) - t_1))
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(N[Log[0.5], $MachinePrecision] * c$95$n), $MachinePrecision]}, N[Exp[N[(N[(N[(N[(c$95$p - c$95$n), $MachinePrecision] * t), $MachinePrecision] * -0.5 + N[(N[(-0.5 * N[(c$95$n - c$95$p), $MachinePrecision]), $MachinePrecision] * s + t$95$1), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 88.4%

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

4. Applied rewrites 93.7%

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

5. Taylor expanded in t around 0

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

7. Applied rewrites 97.3%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 99.3%

    \[\leadsto e^{\mathsf{fma}\left(\left(c\_p - c\_n\right) \cdot t, -0.5, \mathsf{fma}\left(-0.5 \cdot \left(c\_n \cdot 1 - c\_p\right), s, \log 0.5 \cdot c\_n\right)\right) - \color{blue}{\log 0.5 \cdot c\_n}} \]

11. Final simplification 99.3%

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

Alternative 3: 98.6% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

4. Applied rewrites 93.7%

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

5. Taylor expanded in t around 0

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

7. Applied rewrites 97.3%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 99.3%

    \[\leadsto e^{\mathsf{fma}\left(\left(c\_p - c\_n\right) \cdot t, -0.5, \mathsf{fma}\left(-0.5 \cdot \left(c\_n \cdot 1 - c\_p\right), s, \log 0.5 \cdot c\_n\right)\right) - \color{blue}{\log 0.5 \cdot c\_n}} \]

11. Taylor expanded in t around 0

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

13. Applied rewrites 98.5%

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

Alternative 4: 95.3% accurate, 47.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 750000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot c\_n\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 750000000.0) 1.0 (* (* t c_n) 0.5)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 750000000.0) {
		tmp = 1.0;
	} else {
		tmp = (t * c_n) * 0.5;
	}
	return tmp;
}

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) :: tmp
    if (-s <= 750000000.0d0) then
        tmp = 1.0d0
    else
        tmp = (t * c_n) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 750000000.0) {
		tmp = 1.0;
	} else {
		tmp = (t * c_n) * 0.5;
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= 750000000.0:
		tmp = 1.0
	else:
		tmp = (t * c_n) * 0.5
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 750000000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(t * c_n) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= 750000000.0)
		tmp = 1.0;
	else
		tmp = (t * c_n) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 750000000.0], 1.0, N[(N[(t * c$95$n), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 7.5e8

   1. Initial program 89.6%

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

   3. Taylor expanded in 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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 90.1

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 95.4%

      \[\leadsto 1 \]

   if 7.5e8 < (neg.f64 s)

   1. Initial program 55.6%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in c_n around inf

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

   11. Applied rewrites 48.3%

       \[\leadsto \left(t \cdot c\_n\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 95.3% accurate, 49.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -750000000:\\ \;\;\;\;\left(t \cdot c\_n\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(s \cdot c\_p, 0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -750000000.0) (* (* t c_n) 0.5) (fma (* s c_p) 0.5 1.0)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -750000000.0) {
		tmp = (t * c_n) * 0.5;
	} else {
		tmp = fma((s * c_p), 0.5, 1.0);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -750000000.0)
		tmp = Float64(Float64(t * c_n) * 0.5);
	else
		tmp = fma(Float64(s * c_p), 0.5, 1.0);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -750000000.0], N[(N[(t * c$95$n), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(s * c$95$p), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < -7.5e8

   1. Initial program 55.6%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in c_n around inf

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

   11. Applied rewrites 48.3%

       \[\leadsto \left(t \cdot c\_n\right) \cdot 0.5 \]

   if -7.5e8 < s

   1. Initial program 89.6%

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

   3. Taylor expanded in 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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 90.1

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 91.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right), \color{blue}{c\_p}, 1\right) \]

   9. Taylor expanded in s around 0

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

   11. Applied rewrites 93.8%

       \[\leadsto \mathsf{fma}\left(c\_p \cdot s, 0.5, \left(\mathsf{log1p}\left(e^{-t}\right) - \log 2\right) \cdot c\_p\right) + 1 \]

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 95.5%

       \[\leadsto \mathsf{fma}\left(s \cdot c\_p, 0.5, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.3% accurate, 49.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -750000000:\\ \;\;\;\;\left(t \cdot c\_n\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c\_p, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -750000000.0) (* (* t c_n) 0.5) (fma (* t c_p) -0.5 1.0)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -750000000.0) {
		tmp = (t * c_n) * 0.5;
	} else {
		tmp = fma((t * c_p), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -750000000.0)
		tmp = Float64(Float64(t * c_n) * 0.5);
	else
		tmp = fma(Float64(t * c_p), -0.5, 1.0);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -750000000.0], N[(N[(t * c$95$n), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(t * c$95$p), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < -7.5e8

   1. Initial program 55.6%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in c_n around inf

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

   11. Applied rewrites 48.3%

       \[\leadsto \left(t \cdot c\_n\right) \cdot 0.5 \]

   if -7.5e8 < s

   1. Initial program 89.6%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 95.4%

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

   9. Taylor expanded in c_n around 0

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

   11. Applied rewrites 95.4%

       \[\leadsto \mathsf{fma}\left(t \cdot c\_p, -0.5, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.2% accurate, 896.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.4%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
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. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-neg.f64 88.9

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

5. Applied rewrites 88.9%

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

6. Taylor expanded in c_p around 0

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

8. Applied rewrites 92.2%

   \[\leadsto 1 \]
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

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 2024332 
(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))))