Harley's example

Percentage Accurate: 90.7% → 98.7%

Time: 54.9s

Alternatives: 6

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 1.423583389384344e-111
  2. c_n = 1.2755450851285103e-294
  3. t = -1.3447825199485209e-246
  4. s = -9.865300708458809e+219

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: 98.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (log1p (exp (- t)))) (t_2 (log1p (exp (- s)))))
   (*
    (pow (pow (pow (exp c_p) (fma t_2 -1.0 t_1)) 0.5) 2.0)
    (pow
     (pow (/ (- 1.0 (exp (- t_2))) (- 1.0 (exp (- t_1)))) (* 0.5 c_n))
     2.0))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = log1p(exp(-t));
	double t_2 = log1p(exp(-s));
	return pow(pow(pow(exp(c_p), fma(t_2, -1.0, t_1)), 0.5), 2.0) * pow(pow(((1.0 - exp(-t_2)) / (1.0 - exp(-t_1))), (0.5 * c_n)), 2.0);
}

Julia

function code(c_p, c_n, t, s)
	t_1 = log1p(exp(Float64(-t)))
	t_2 = log1p(exp(Float64(-s)))
	return Float64((((exp(c_p) ^ fma(t_2, -1.0, t_1)) ^ 0.5) ^ 2.0) * ((Float64(Float64(1.0 - exp(Float64(-t_2))) / Float64(1.0 - exp(Float64(-t_1)))) ^ Float64(0.5 * c_n)) ^ 2.0))
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Log[1 + N[Exp[(-s)], $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[Power[N[Power[N[Exp[c$95$p], $MachinePrecision], N[(t$95$2 * -1.0 + t$95$1), $MachinePrecision]], $MachinePrecision], 0.5], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[Power[N[(N[(1.0 - N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * c$95$n), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 91.5%

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

4. Applied rewrites 97.1%

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

6. Applied rewrites 99.0%

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

7. Final simplification 99.0%

   \[\leadsto {\left({\left({\left(e^{c\_p}\right)}^{\left(\mathsf{fma}\left(\mathsf{log1p}\left(e^{-s}\right), -1, \mathsf{log1p}\left(e^{-t}\right)\right)\right)}\right)}^{0.5}\right)}^{2} \cdot {\left({\left(\frac{1 - e^{-\mathsf{log1p}\left(e^{-s}\right)}}{1 - e^{-\mathsf{log1p}\left(e^{-t}\right)}}\right)}^{\left(0.5 \cdot c\_n\right)}\right)}^{2} \]
8. Add Preprocessing

Alternative 2: 98.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- t))) (t_2 (exp (- s))))
   (*
    (pow
     (pow
      (exp (- (log1p (/ -1.0 (+ t_2 1.0))) (log1p (/ -1.0 (+ t_1 1.0)))))
      2.0)
     (* 0.5 c_n))
    (pow (exp c_p) (fma (log1p t_2) -1.0 (log1p t_1))))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-t);
	double t_2 = exp(-s);
	return pow(pow(exp((log1p((-1.0 / (t_2 + 1.0))) - log1p((-1.0 / (t_1 + 1.0))))), 2.0), (0.5 * c_n)) * pow(exp(c_p), fma(log1p(t_2), -1.0, log1p(t_1)));
}

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-t))
	t_2 = exp(Float64(-s))
	return Float64(((exp(Float64(log1p(Float64(-1.0 / Float64(t_2 + 1.0))) - log1p(Float64(-1.0 / Float64(t_1 + 1.0))))) ^ 2.0) ^ Float64(0.5 * c_n)) * (exp(c_p) ^ fma(log1p(t_2), -1.0, log1p(t_1))))
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[(-t)], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-s)], $MachinePrecision]}, N[(N[Power[N[Power[N[Exp[N[(N[Log[1 + N[(-1.0 / N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(-1.0 / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], N[(0.5 * c$95$n), $MachinePrecision]], $MachinePrecision] * N[Power[N[Exp[c$95$p], $MachinePrecision], N[(N[Log[1 + t$95$2], $MachinePrecision] * -1.0 + N[Log[1 + t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 91.5%

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

4. Applied rewrites 97.1%

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

6. Applied rewrites 99.0%

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

8. Applied rewrites 99.0%

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

9. Final simplification 99.0%

   \[\leadsto {\left({\left(e^{\mathsf{log1p}\left(\frac{-1}{e^{-s} + 1}\right) - \mathsf{log1p}\left(\frac{-1}{e^{-t} + 1}\right)}\right)}^{2}\right)}^{\left(0.5 \cdot c\_n\right)} \cdot {\left(e^{c\_p}\right)}^{\left(\mathsf{fma}\left(\mathsf{log1p}\left(e^{-s}\right), -1, \mathsf{log1p}\left(e^{-t}\right)\right)\right)} \]
10. Add Preprocessing

Alternative 3: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{log1p}\left(e^{-s}\right)\\ t_2 := \mathsf{log1p}\left(e^{-t}\right)\\ {\left(e^{c\_p}\right)}^{\left(\mathsf{fma}\left(t\_1, -1, t\_2\right)\right)} \cdot {\left(\frac{1 - e^{-t\_1}}{1 - e^{-t\_2}}\right)}^{c\_n} \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (log1p (exp (- s)))) (t_2 (log1p (exp (- t)))))
   (*
    (pow (exp c_p) (fma t_1 -1.0 t_2))
    (pow (/ (- 1.0 (exp (- t_1))) (- 1.0 (exp (- t_2)))) c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = log1p(exp(-s));
	double t_2 = log1p(exp(-t));
	return pow(exp(c_p), fma(t_1, -1.0, t_2)) * pow(((1.0 - exp(-t_1)) / (1.0 - exp(-t_2))), c_n);
}

Julia

function code(c_p, c_n, t, s)
	t_1 = log1p(exp(Float64(-s)))
	t_2 = log1p(exp(Float64(-t)))
	return Float64((exp(c_p) ^ fma(t_1, -1.0, t_2)) * (Float64(Float64(1.0 - exp(Float64(-t_1))) / Float64(1.0 - exp(Float64(-t_2)))) ^ c_n))
end

Wolfram

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

Derivation

1. Initial program 91.5%

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

4. Applied rewrites 97.1%

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

6. Applied rewrites 99.0%

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

7. Final simplification 99.0%

   \[\leadsto {\left(e^{c\_p}\right)}^{\left(\mathsf{fma}\left(\mathsf{log1p}\left(e^{-s}\right), -1, \mathsf{log1p}\left(e^{-t}\right)\right)\right)} \cdot {\left(\frac{1 - e^{-\mathsf{log1p}\left(e^{-s}\right)}}{1 - e^{-\mathsf{log1p}\left(e^{-t}\right)}}\right)}^{c\_n} \]
8. Add Preprocessing

Alternative 4: 98.5% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

4. Applied rewrites 97.1%

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

5. Taylor expanded in t around 0

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

   1. lower-exp.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 97.1%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 98.7%

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

11. Final simplification 98.7%

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

Alternative 5: 98.5% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

4. Applied rewrites 97.1%

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

5. Taylor expanded in t around 0

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

   1. lower-exp.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 97.1%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 98.6%

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

11. Final simplification 98.6%

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

Alternative 6: 93.9% 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 91.5%

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

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

   6. lower-+.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. neg-mul-1 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-exp.f64 N/A

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

   15. lower-neg.f64 95.6

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

5. Applied rewrites 95.6%

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

6. Taylor expanded in c_p around 0

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

8. Applied rewrites 96.2%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer Target 1: 96.2% 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 2024242 
(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))))