Harley's example

Percentage Accurate: 91.1% → 99.6%

Time: 54.0s

Alternatives: 6

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 5.844155452492486e-223
  2. c_n = 2.785937089773872e-280
  3. t = 1.921854018997517e+122
  4. s = 1.4261867845450986e-80

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.1% 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: 99.6% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.7%

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

4. Applied rewrites 96.1%

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

5. Taylor expanded in s around 0

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

   1. lower-fma.f64 N/A

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

7. Applied rewrites 97.0%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(c\_n, \log 0.5 - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right), \mathsf{fma}\left(c\_p, \mathsf{log1p}\left(e^{-t}\right) - \log 2, s \cdot \mathsf{fma}\left(s, -0.125 \cdot \left(c\_n + c\_p\right), \mathsf{fma}\left(c\_n, -0.5, c\_p \cdot 0.5\right)\right)\right)\right)}} \]

8. Taylor expanded in t around 0

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

10. Applied rewrites 99.2%

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

Alternative 2: 98.7% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.7%

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

4. Applied rewrites 96.1%

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

5. Taylor expanded in s around 0

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

   1. lower-fma.f64 N/A

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

7. Applied rewrites 97.0%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(c\_n, \log 0.5 - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right), \mathsf{fma}\left(c\_p, \mathsf{log1p}\left(e^{-t}\right) - \log 2, s \cdot \mathsf{fma}\left(s, -0.125 \cdot \left(c\_n + c\_p\right), \mathsf{fma}\left(c\_n, -0.5, c\_p \cdot 0.5\right)\right)\right)\right)}} \]

8. Taylor expanded in t around 0

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

10. Applied rewrites 98.5%

    \[\leadsto e^{s \cdot \color{blue}{\mathsf{fma}\left(c\_n, \mathsf{fma}\left(-0.125, s, -0.5\right), c\_p \cdot \mathsf{fma}\left(-0.125, s, 0.5\right)\right)}} \]
11. Add Preprocessing

Alternative 3: 98.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ e^{\left(-0.125 \cdot \left(c\_p + c\_n\right)\right) \cdot \left(s \cdot s\right)} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = exp((((-0.125d0) * (c_p + c_n)) * (s * s)))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp(((-0.125 * (c_p + c_n)) * (s * s)));
}

Python

def code(c_p, c_n, t, s):
	return math.exp(((-0.125 * (c_p + c_n)) * (s * s)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.7%

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

4. Applied rewrites 96.1%

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

5. Taylor expanded in s around 0

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

   1. lower-fma.f64 N/A

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

7. Applied rewrites 97.0%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(c\_n, \log 0.5 - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right), \mathsf{fma}\left(c\_p, \mathsf{log1p}\left(e^{-t}\right) - \log 2, s \cdot \mathsf{fma}\left(s, -0.125 \cdot \left(c\_n + c\_p\right), \mathsf{fma}\left(c\_n, -0.5, c\_p \cdot 0.5\right)\right)\right)\right)}} \]

8. Taylor expanded in s around inf

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

10. Applied rewrites 98.1%

    \[\leadsto e^{\left(-0.125 \cdot \left(c\_n + c\_p\right)\right) \cdot \color{blue}{\left(s \cdot s\right)}} \]

11. Final simplification 98.1%

    \[\leadsto e^{\left(-0.125 \cdot \left(c\_p + c\_n\right)\right) \cdot \left(s \cdot s\right)} \]
12. Add Preprocessing

Alternative 4: 98.1% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.7%

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

4. Applied rewrites 96.1%

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

5. Taylor expanded in s around 0

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

   1. lower-fma.f64 N/A

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

7. Applied rewrites 97.0%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(c\_n, \log 0.5 - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right), \mathsf{fma}\left(c\_p, \mathsf{log1p}\left(e^{-t}\right) - \log 2, s \cdot \mathsf{fma}\left(s, -0.125 \cdot \left(c\_n + c\_p\right), \mathsf{fma}\left(c\_n, -0.5, c\_p \cdot 0.5\right)\right)\right)\right)}} \]

8. Taylor expanded in t around 0

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

10. Applied rewrites 98.5%

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

11. Taylor expanded in c_n around 0

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

13. Applied rewrites 97.8%

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

14. Final simplification 97.8%

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

Alternative 5: 98.1% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.7%

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

4. Applied rewrites 96.1%

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

5. Taylor expanded in s around 0

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

   1. lower-fma.f64 N/A

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

7. Applied rewrites 97.0%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(c\_n, \log 0.5 - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right), \mathsf{fma}\left(c\_p, \mathsf{log1p}\left(e^{-t}\right) - \log 2, s \cdot \mathsf{fma}\left(s, -0.125 \cdot \left(c\_n + c\_p\right), \mathsf{fma}\left(c\_n, -0.5, c\_p \cdot 0.5\right)\right)\right)\right)}} \]

8. Taylor expanded in t around 0

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

10. Applied rewrites 98.5%

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

11. Taylor expanded in c_n around inf

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

13. Applied rewrites 97.3%

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

14. Final simplification 97.3%

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

Alternative 6: 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 92.7%

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

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 95.4

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

5. Applied rewrites 95.4%

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

6. Taylor expanded in c_p around 0

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

8. Applied rewrites 95.5%

   \[\leadsto 1 \]
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 2024226 
(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))))