peicewise-defined

Percentage Accurate: 76.6% → 77.3%

Time: 5.1s

Alternatives: 7

Speedup: 1.5×

Specification

\[p \geq -1.79 \cdot 10^{+308} \land p \leq 1.79 \cdot 10^{+308}\]

Math

\[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]

FPCore

(FPCore (p)
  :precision binary64
  (- (+ p 1.0) (/ (pow (fmin p 0.0) 2.0) (- (fmin p 0.0) 1.0))))

C

double code(double p) {
	return (p + 1.0) - (pow(fmin(p, 0.0), 2.0) / (fmin(p, 0.0) - 1.0));
}

Fortran

real(8) function code(p)
use fmin_fmax_functions
    real(8), intent (in) :: p
    code = (p + 1.0d0) - ((fmin(p, 0.0d0) ** 2.0d0) / (fmin(p, 0.0d0) - 1.0d0))
end function

Java

public static double code(double p) {
	return (p + 1.0) - (Math.pow(fmin(p, 0.0), 2.0) / (fmin(p, 0.0) - 1.0));
}

Python

def code(p):
	return (p + 1.0) - (math.pow(fmin(p, 0.0), 2.0) / (fmin(p, 0.0) - 1.0))

Julia

function code(p)
	return Float64(Float64(p + 1.0) - Float64((fmin(p, 0.0) ^ 2.0) / Float64(fmin(p, 0.0) - 1.0)))
end

MATLAB

function tmp = code(p)
	tmp = (p + 1.0) - ((min(p, 0.0) ^ 2.0) / (min(p, 0.0) - 1.0));
end

Wolfram

code[p_] := N[(N[(p + 1.0), $MachinePrecision] - N[(N[Power[N[Min[p, 0.0], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Min[p, 0.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.6% accurate, 1.0× speedup

Math

\[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]

FPCore

(FPCore (p)
  :precision binary64
  (- (+ p 1.0) (/ (pow (fmin p 0.0) 2.0) (- (fmin p 0.0) 1.0))))

C

double code(double p) {
	return (p + 1.0) - (pow(fmin(p, 0.0), 2.0) / (fmin(p, 0.0) - 1.0));
}

Fortran

real(8) function code(p)
use fmin_fmax_functions
    real(8), intent (in) :: p
    code = (p + 1.0d0) - ((fmin(p, 0.0d0) ** 2.0d0) / (fmin(p, 0.0d0) - 1.0d0))
end function

Java

public static double code(double p) {
	return (p + 1.0) - (Math.pow(fmin(p, 0.0), 2.0) / (fmin(p, 0.0) - 1.0));
}

Python

def code(p):
	return (p + 1.0) - (math.pow(fmin(p, 0.0), 2.0) / (fmin(p, 0.0) - 1.0))

Julia

function code(p)
	return Float64(Float64(p + 1.0) - Float64((fmin(p, 0.0) ^ 2.0) / Float64(fmin(p, 0.0) - 1.0)))
end

MATLAB

function tmp = code(p)
	tmp = (p + 1.0) - ((min(p, 0.0) ^ 2.0) / (min(p, 0.0) - 1.0));
end

Wolfram

code[p_] := N[(N[(p + 1.0), $MachinePrecision] - N[(N[Power[N[Min[p, 0.0], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Min[p, 0.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 77.3% accurate, 1.4× speedup

Math

\[-\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1}, -1 - p\right) \]

FPCore

(FPCore (p)
  :precision binary64
  (-
 (fma (fmin 0.0 p) (/ (fmin 0.0 p) (- (fmin 0.0 p) 1.0)) (- -1.0 p))))

C

double code(double p) {
	return -fma(fmin(0.0, p), (fmin(0.0, p) / (fmin(0.0, p) - 1.0)), (-1.0 - p));
}

Julia

function code(p)
	return Float64(-fma(fmin(0.0, p), Float64(fmin(0.0, p) / Float64(fmin(0.0, p) - 1.0)), Float64(-1.0 - p)))
end

Wolfram

code[p_] := (-N[(N[Min[0.0, p], $MachinePrecision] * N[(N[Min[0.0, p], $MachinePrecision] / N[(N[Min[0.0, p], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - p), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 76.6%

   \[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

   2. sub-negate-rev N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} - \left(p + 1\right)\right)\right)} \]

   3. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} - \left(p + 1\right)\right)} \]

   4. sub-flip N/A

      \[\leadsto -\color{blue}{\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} + \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right)\right)} \]

   5. lift-/.f64 N/A

      \[\leadsto -\left(\color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} + \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right)\right) \]

   6. lift-pow.f64 N/A

      \[\leadsto -\left(\frac{\color{blue}{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}}{\mathsf{min}\left(p, 0\right) - 1} + \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto -\left(\frac{\color{blue}{\mathsf{min}\left(p, 0\right) \cdot \mathsf{min}\left(p, 0\right)}}{\mathsf{min}\left(p, 0\right) - 1} + \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto -\left(\color{blue}{\mathsf{min}\left(p, 0\right) \cdot \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}} + \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto -\color{blue}{\mathsf{fma}\left(\mathsf{min}\left(p, 0\right), \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}, \mathsf{neg}\left(\left(p + 1\right)\right)\right)} \]

   10. lift-fmin.f64 N/A

       \[\leadsto -\mathsf{fma}\left(\color{blue}{\mathsf{min}\left(p, 0\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}, \mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

   11. fmin-swap N/A

       \[\leadsto -\mathsf{fma}\left(\color{blue}{\mathsf{min}\left(0, p\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}, \mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

   12. lower-fmin.f64 N/A

       \[\leadsto -\mathsf{fma}\left(\color{blue}{\mathsf{min}\left(0, p\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}, \mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

   13. lower-/.f64 N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \color{blue}{\frac{\mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}}, \mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

   14. lift-fmin.f64 N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(p, 0\right)}}{\mathsf{min}\left(p, 0\right) - 1}, \mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

   15. fmin-swap N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{min}\left(p, 0\right) - 1}, \mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

   16. lower-fmin.f64 N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{min}\left(p, 0\right) - 1}, \mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

   17. lift-fmin.f64 N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{\mathsf{min}\left(p, 0\right)} - 1}, \mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

   18. fmin-swap N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1}, \mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

   19. lower-fmin.f64 N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1}, \mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

   20. lift-+.f64 N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1}, \mathsf{neg}\left(\color{blue}{\left(p + 1\right)}\right)\right) \]

   21. add-flip N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1}, \mathsf{neg}\left(\color{blue}{\left(p - \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

   22. sub-negate-rev N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1}, \color{blue}{\left(\mathsf{neg}\left(1\right)\right) - p}\right) \]

   23. lower--.f64 N/A

       \[\leadsto -\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1}, \color{blue}{\left(\mathsf{neg}\left(1\right)\right) - p}\right) \]

   24. metadata-eval 77.3%

       \[\leadsto -\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1}, \color{blue}{-1} - p\right) \]

3. Applied rewrites 77.3%

   \[\leadsto \color{blue}{-\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1}, -1 - p\right)} \]
4. Add Preprocessing

Alternative 2: 77.2% accurate, 1.5× speedup

Math

\[\mathsf{fma}\left(\frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \mathsf{min}\left(0, p\right), p\right) - -1 \]

FPCore

(FPCore (p)
  :precision binary64
  (- (fma (/ (fmin 0.0 p) (- 1.0 (fmin 0.0 p))) (fmin 0.0 p) p) -1.0))

C

double code(double p) {
	return fma((fmin(0.0, p) / (1.0 - fmin(0.0, p))), fmin(0.0, p), p) - -1.0;
}

Julia

function code(p)
	return Float64(fma(Float64(fmin(0.0, p) / Float64(1.0 - fmin(0.0, p))), fmin(0.0, p), p) - -1.0)
end

Wolfram

code[p_] := N[(N[(N[(N[Min[0.0, p], $MachinePrecision] / N[(1.0 - N[Min[0.0, p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Min[0.0, p], $MachinePrecision] + p), $MachinePrecision] - -1.0), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

   2. lift-/.f64 N/A

      \[\leadsto \left(p + 1\right) - \color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

   3. frac-2neg N/A

      \[\leadsto \left(p + 1\right) - \color{blue}{\frac{\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

   4. sub-to-fraction N/A

      \[\leadsto \color{blue}{\frac{\left(p + 1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right) - \left(\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

   5. mult-flip N/A

      \[\leadsto \color{blue}{\left(\left(p + 1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right) - \left(\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(p + 1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right) - \left(\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

3. Applied rewrites 76.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(p - -1, 1 - \mathsf{min}\left(0, p\right), \mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)\right) \cdot \frac{-1}{\mathsf{min}\left(0, p\right) - 1}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(p - -1, 1 - \mathsf{min}\left(0, p\right), \mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)\right) \cdot \frac{-1}{\mathsf{min}\left(0, p\right) - 1}} \]

   2. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\left(\left(p - -1\right) \cdot \left(1 - \mathsf{min}\left(0, p\right)\right) + \mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)\right)} \cdot \frac{-1}{\mathsf{min}\left(0, p\right) - 1} \]

   3. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(1 - \mathsf{min}\left(0, p\right)\right) \cdot \left(p - -1\right)} + \mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)\right) \cdot \frac{-1}{\mathsf{min}\left(0, p\right) - 1} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(1 - \mathsf{min}\left(0, p\right)\right) \cdot \left(p - -1\right)} + \mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)\right) \cdot \frac{-1}{\mathsf{min}\left(0, p\right) - 1} \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right) + \left(1 - \mathsf{min}\left(0, p\right)\right) \cdot \left(p - -1\right)\right)} \cdot \frac{-1}{\mathsf{min}\left(0, p\right) - 1} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)} + \left(1 - \mathsf{min}\left(0, p\right)\right) \cdot \left(p - -1\right)\right) \cdot \frac{-1}{\mathsf{min}\left(0, p\right) - 1} \]

   7. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \mathsf{min}\left(0, p\right), \left(1 - \mathsf{min}\left(0, p\right)\right) \cdot \left(p - -1\right)\right)} \cdot \frac{-1}{\mathsf{min}\left(0, p\right) - 1} \]

   8. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \mathsf{min}\left(0, p\right), \left(1 - \mathsf{min}\left(0, p\right)\right) \cdot \left(p - -1\right)\right) \cdot \color{blue}{\frac{-1}{\mathsf{min}\left(0, p\right) - 1}} \]

   9. frac-2neg N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \mathsf{min}\left(0, p\right), \left(1 - \mathsf{min}\left(0, p\right)\right) \cdot \left(p - -1\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(0, p\right) - 1\right)\right)}} \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \mathsf{min}\left(0, p\right), \left(1 - \mathsf{min}\left(0, p\right)\right) \cdot \left(p - -1\right)\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(\left(\mathsf{min}\left(0, p\right) - 1\right)\right)} \]

   11. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \mathsf{min}\left(0, p\right), \left(1 - \mathsf{min}\left(0, p\right)\right) \cdot \left(p - -1\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{min}\left(0, p\right) - 1\right)}\right)} \]

   12. sub-negate-rev N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \mathsf{min}\left(0, p\right), \left(1 - \mathsf{min}\left(0, p\right)\right) \cdot \left(p - -1\right)\right) \cdot \frac{1}{\color{blue}{1 - \mathsf{min}\left(0, p\right)}} \]

   13. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \mathsf{min}\left(0, p\right), \left(1 - \mathsf{min}\left(0, p\right)\right) \cdot \left(p - -1\right)\right) \cdot \frac{1}{\color{blue}{1 - \mathsf{min}\left(0, p\right)}} \]

   14. mult-flip N/A

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{min}\left(0, p\right), \mathsf{min}\left(0, p\right), \left(1 - \mathsf{min}\left(0, p\right)\right) \cdot \left(p - -1\right)\right)}{1 - \mathsf{min}\left(0, p\right)}} \]

5. Applied rewrites 77.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}, \mathsf{min}\left(0, p\right), p\right) - -1} \]
6. Add Preprocessing

Alternative 3: 77.2% accurate, 1.5× speedup

Math

\[p - \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1}, -1\right) \]

FPCore

(FPCore (p)
  :precision binary64
  (- p (fma (fmin 0.0 p) (/ (fmin 0.0 p) (- (fmin 0.0 p) 1.0)) -1.0)))

C

double code(double p) {
	return p - fma(fmin(0.0, p), (fmin(0.0, p) / (fmin(0.0, p) - 1.0)), -1.0);
}

Julia

function code(p)
	return Float64(p - fma(fmin(0.0, p), Float64(fmin(0.0, p) / Float64(fmin(0.0, p) - 1.0)), -1.0))
end

Wolfram

code[p_] := N[(p - N[(N[Min[0.0, p], $MachinePrecision] * N[(N[Min[0.0, p], $MachinePrecision] / N[(N[Min[0.0, p], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(p + 1\right)} - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]

   3. associate--l+ N/A

      \[\leadsto \color{blue}{p + \left(1 - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)} \]

   4. sub-negate-rev N/A

      \[\leadsto p + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} - 1\right)\right)\right)} \]

   5. sub-flip-reverse N/A

      \[\leadsto \color{blue}{p - \left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} - 1\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \color{blue}{p - \left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} - 1\right)} \]

   7. sub-flip N/A

      \[\leadsto p - \color{blue}{\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   8. lift-/.f64 N/A

      \[\leadsto p - \left(\color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   9. lift-pow.f64 N/A

      \[\leadsto p - \left(\frac{\color{blue}{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}}{\mathsf{min}\left(p, 0\right) - 1} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto p - \left(\frac{\color{blue}{\mathsf{min}\left(p, 0\right) \cdot \mathsf{min}\left(p, 0\right)}}{\mathsf{min}\left(p, 0\right) - 1} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   11. associate-/l* N/A

       \[\leadsto p - \left(\color{blue}{\mathsf{min}\left(p, 0\right) \cdot \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto p - \color{blue}{\mathsf{fma}\left(\mathsf{min}\left(p, 0\right), \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}, \mathsf{neg}\left(1\right)\right)} \]

   13. lift-fmin.f64 N/A

       \[\leadsto p - \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(p, 0\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}, \mathsf{neg}\left(1\right)\right) \]

   14. fmin-swap N/A

       \[\leadsto p - \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(0, p\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}, \mathsf{neg}\left(1\right)\right) \]

   15. lower-fmin.f64 N/A

       \[\leadsto p - \mathsf{fma}\left(\color{blue}{\mathsf{min}\left(0, p\right)}, \frac{\mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}, \mathsf{neg}\left(1\right)\right) \]

   16. lower-/.f64 N/A

       \[\leadsto p - \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \color{blue}{\frac{\mathsf{min}\left(p, 0\right)}{\mathsf{min}\left(p, 0\right) - 1}}, \mathsf{neg}\left(1\right)\right) \]

   17. lift-fmin.f64 N/A

       \[\leadsto p - \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(p, 0\right)}}{\mathsf{min}\left(p, 0\right) - 1}, \mathsf{neg}\left(1\right)\right) \]

   18. fmin-swap N/A

       \[\leadsto p - \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{min}\left(p, 0\right) - 1}, \mathsf{neg}\left(1\right)\right) \]

   19. lower-fmin.f64 N/A

       \[\leadsto p - \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{min}\left(p, 0\right) - 1}, \mathsf{neg}\left(1\right)\right) \]

   20. lift-fmin.f64 N/A

       \[\leadsto p - \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{\mathsf{min}\left(p, 0\right)} - 1}, \mathsf{neg}\left(1\right)\right) \]

   21. fmin-swap N/A

       \[\leadsto p - \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1}, \mathsf{neg}\left(1\right)\right) \]

   22. lower-fmin.f64 N/A

       \[\leadsto p - \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1}, \mathsf{neg}\left(1\right)\right) \]

   23. metadata-eval 77.2%

       \[\leadsto p - \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1}, \color{blue}{-1}\right) \]

3. Applied rewrites 77.2%

   \[\leadsto \color{blue}{p - \mathsf{fma}\left(\mathsf{min}\left(0, p\right), \frac{\mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1}, -1\right)} \]
4. Add Preprocessing

Alternative 4: 74.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(0, p\right) - 1\\ \mathbf{if}\;p \leq 600000000000:\\ \;\;\;\;\mathsf{min}\left(0, 0\right) \cdot \frac{\mathsf{min}\left(0, 0\right)}{1 - \mathsf{min}\left(0, 0\right)} - -1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot p}{t\_0}\\ \end{array} \]

FPCore

(FPCore (p)
  :precision binary64
  (let* ((t_0 (- (fmin 0.0 p) 1.0)))
  (if (<= p 600000000000.0)
    (-
     (* (fmin 0.0 0.0) (/ (fmin 0.0 0.0) (- 1.0 (fmin 0.0 0.0))))
     -1.0)
    (/ (* t_0 p) t_0))))

C

double code(double p) {
	double t_0 = fmin(0.0, p) - 1.0;
	double tmp;
	if (p <= 600000000000.0) {
		tmp = (fmin(0.0, 0.0) * (fmin(0.0, 0.0) / (1.0 - fmin(0.0, 0.0)))) - -1.0;
	} else {
		tmp = (t_0 * p) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(p)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8) :: t_0
    real(8) :: tmp
    t_0 = fmin(0.0d0, p) - 1.0d0
    if (p <= 600000000000.0d0) then
        tmp = (fmin(0.0d0, 0.0d0) * (fmin(0.0d0, 0.0d0) / (1.0d0 - fmin(0.0d0, 0.0d0)))) - (-1.0d0)
    else
        tmp = (t_0 * p) / t_0
    end if
    code = tmp
end function

Java

public static double code(double p) {
	double t_0 = fmin(0.0, p) - 1.0;
	double tmp;
	if (p <= 600000000000.0) {
		tmp = (fmin(0.0, 0.0) * (fmin(0.0, 0.0) / (1.0 - fmin(0.0, 0.0)))) - -1.0;
	} else {
		tmp = (t_0 * p) / t_0;
	}
	return tmp;
}

Python

def code(p):
	t_0 = fmin(0.0, p) - 1.0
	tmp = 0
	if p <= 600000000000.0:
		tmp = (fmin(0.0, 0.0) * (fmin(0.0, 0.0) / (1.0 - fmin(0.0, 0.0)))) - -1.0
	else:
		tmp = (t_0 * p) / t_0
	return tmp

Julia

function code(p)
	t_0 = Float64(fmin(0.0, p) - 1.0)
	tmp = 0.0
	if (p <= 600000000000.0)
		tmp = Float64(Float64(fmin(0.0, 0.0) * Float64(fmin(0.0, 0.0) / Float64(1.0 - fmin(0.0, 0.0)))) - -1.0);
	else
		tmp = Float64(Float64(t_0 * p) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p)
	t_0 = min(0.0, p) - 1.0;
	tmp = 0.0;
	if (p <= 600000000000.0)
		tmp = (min(0.0, 0.0) * (min(0.0, 0.0) / (1.0 - min(0.0, 0.0)))) - -1.0;
	else
		tmp = (t_0 * p) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_] := Block[{t$95$0 = N[(N[Min[0.0, p], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[p, 600000000000.0], N[(N[(N[Min[0.0, 0.0], $MachinePrecision] * N[(N[Min[0.0, 0.0], $MachinePrecision] / N[(1.0 - N[Min[0.0, 0.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision], N[(N[(t$95$0 * p), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if p < 6e11

   1. Initial program 76.6%

      \[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

      2. sub-flip N/A

         \[\leadsto \color{blue}{\left(p + 1\right) + \left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right) + \left(p + 1\right)} \]

      4. add-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right) - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right) - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}}\right)\right) - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      7. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{min}\left(p, 0\right) \cdot \mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{min}\left(p, 0\right) \cdot \mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      12. lift-fmin.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{min}\left(p, 0\right)} \cdot \mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      13. fmin-swap N/A

          \[\leadsto \frac{\color{blue}{\mathsf{min}\left(0, p\right)} \cdot \mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      14. lower-fmin.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{min}\left(0, p\right)} \cdot \mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      15. lift-fmin.f64 N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \color{blue}{\mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      16. fmin-swap N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      17. lower-fmin.f64 N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      18. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{min}\left(p, 0\right) - 1\right)}\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      19. sub-negate-rev N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{\color{blue}{1 - \mathsf{min}\left(p, 0\right)}} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      20. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{\color{blue}{1 - \mathsf{min}\left(p, 0\right)}} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      21. lift-fmin.f64 N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(p, 0\right)}} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      22. fmin-swap N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(0, p\right)}} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      23. lower-fmin.f64 N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(0, p\right)}} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      24. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)} - \left(\mathsf{neg}\left(\color{blue}{\left(p + 1\right)}\right)\right) \]

      25. add-flip N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)} - \left(\mathsf{neg}\left(\color{blue}{\left(p - \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

   3. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)} - \left(-1 - p\right)} \]

   4. Taylor expanded in p around 0

      \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)} - \color{blue}{-1} \]
   5. Step-by-step derivation

   6. Applied rewrites 51.9%

      \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)} - \color{blue}{-1} \]

   7. Taylor expanded in undef-var around zero

      \[\leadsto \frac{\mathsf{min}\left(0, \color{blue}{0}\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)} - -1 \]
   8. Step-by-step derivation

   9. Applied rewrites 52.3%

      \[\leadsto \frac{\mathsf{min}\left(0, \color{blue}{0}\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)} - -1 \]

   10. Taylor expanded in undef-var around zero

       \[\leadsto \frac{\mathsf{min}\left(0, 0\right) \cdot \mathsf{min}\left(0, \color{blue}{0}\right)}{1 - \mathsf{min}\left(0, p\right)} - -1 \]
   11. Step-by-step derivation

   12. Applied rewrites 52.3%

       \[\leadsto \frac{\mathsf{min}\left(0, 0\right) \cdot \mathsf{min}\left(0, \color{blue}{0}\right)}{1 - \mathsf{min}\left(0, p\right)} - -1 \]

   13. Taylor expanded in undef-var around zero

       \[\leadsto \frac{\mathsf{min}\left(0, 0\right) \cdot \mathsf{min}\left(0, 0\right)}{1 - \mathsf{min}\left(0, \color{blue}{0}\right)} - -1 \]
   14. Step-by-step derivation

   15. Applied rewrites 52.3%

       \[\leadsto \frac{\mathsf{min}\left(0, 0\right) \cdot \mathsf{min}\left(0, 0\right)}{1 - \mathsf{min}\left(0, \color{blue}{0}\right)} - -1 \]
   16. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{min}\left(0, 0\right) \cdot \mathsf{min}\left(0, 0\right)}{1 - \mathsf{min}\left(0, 0\right)}} - -1 \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{min}\left(0, 0\right) \cdot \mathsf{min}\left(0, 0\right)}}{1 - \mathsf{min}\left(0, 0\right)} - -1 \]

       3. associate-/l* N/A

          \[\leadsto \color{blue}{\mathsf{min}\left(0, 0\right) \cdot \frac{\mathsf{min}\left(0, 0\right)}{1 - \mathsf{min}\left(0, 0\right)}} - -1 \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\mathsf{min}\left(0, 0\right) \cdot \frac{\mathsf{min}\left(0, 0\right)}{1 - \mathsf{min}\left(0, 0\right)}} - -1 \]

       5. lift-fmin.f64 N/A

          \[\leadsto \color{blue}{\mathsf{min}\left(0, 0\right)} \cdot \frac{\mathsf{min}\left(0, 0\right)}{1 - \mathsf{min}\left(0, 0\right)} - -1 \]

       6. fmin-swap N/A

          \[\leadsto \color{blue}{\mathsf{min}\left(0, 0\right)} \cdot \frac{\mathsf{min}\left(0, 0\right)}{1 - \mathsf{min}\left(0, 0\right)} - -1 \]

       7. lower-fmin.f64 N/A

          \[\leadsto \color{blue}{\mathsf{min}\left(0, 0\right)} \cdot \frac{\mathsf{min}\left(0, 0\right)}{1 - \mathsf{min}\left(0, 0\right)} - -1 \]

       8. lower-/.f64 52.3%

          \[\leadsto \mathsf{min}\left(0, 0\right) \cdot \color{blue}{\frac{\mathsf{min}\left(0, 0\right)}{1 - \mathsf{min}\left(0, 0\right)}} - -1 \]

       9. lift-fmin.f64 N/A

          \[\leadsto \mathsf{min}\left(0, 0\right) \cdot \frac{\color{blue}{\mathsf{min}\left(0, 0\right)}}{1 - \mathsf{min}\left(0, 0\right)} - -1 \]

       10. fmin-swap N/A

           \[\leadsto \mathsf{min}\left(0, 0\right) \cdot \frac{\color{blue}{\mathsf{min}\left(0, 0\right)}}{1 - \mathsf{min}\left(0, 0\right)} - -1 \]

       11. lower-fmin.f64 52.3%

           \[\leadsto \mathsf{min}\left(0, 0\right) \cdot \frac{\color{blue}{\mathsf{min}\left(0, 0\right)}}{1 - \mathsf{min}\left(0, 0\right)} - -1 \]

       12. lift-fmin.f64 N/A

           \[\leadsto \mathsf{min}\left(0, 0\right) \cdot \frac{\mathsf{min}\left(0, 0\right)}{1 - \color{blue}{\mathsf{min}\left(0, 0\right)}} - -1 \]

       13. fmin-swap N/A

           \[\leadsto \mathsf{min}\left(0, 0\right) \cdot \frac{\mathsf{min}\left(0, 0\right)}{1 - \color{blue}{\mathsf{min}\left(0, 0\right)}} - -1 \]

       14. lower-fmin.f64 52.3%

           \[\leadsto \mathsf{min}\left(0, 0\right) \cdot \frac{\mathsf{min}\left(0, 0\right)}{1 - \color{blue}{\mathsf{min}\left(0, 0\right)}} - -1 \]

   17. Applied rewrites 52.3%

       \[\leadsto \color{blue}{\mathsf{min}\left(0, 0\right) \cdot \frac{\mathsf{min}\left(0, 0\right)}{1 - \mathsf{min}\left(0, 0\right)}} - -1 \]

   if 6e11 < p

   1. Initial program 76.6%

      \[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(p + 1\right) - \color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

      3. frac-2neg N/A

         \[\leadsto \left(p + 1\right) - \color{blue}{\frac{\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

      4. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(p + 1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right) - \left(\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

      5. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(p + 1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right) - \left(\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(p + 1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right) - \left(\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

   3. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(p - -1, 1 - \mathsf{min}\left(0, p\right), \mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)\right) \cdot \frac{-1}{\mathsf{min}\left(0, p\right) - 1}} \]

   4. Taylor expanded in p around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1} \]

      4. lower--.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, \color{blue}{p}\right) - 1} \]

      5. lower-fmin.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1} \]

      6. lower--.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - \color{blue}{1}} \]

      7. lower-fmin.f64 26.5%

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1} \]

   6. Applied rewrites 26.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)\right)}{\mathsf{min}\left(\color{blue}{0}, p\right) - 1} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 - \mathsf{min}\left(0, p\right)\right) \cdot p\right)}{\mathsf{min}\left(\color{blue}{0}, p\right) - 1} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - \mathsf{min}\left(0, p\right)\right)\right)\right) \cdot p}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - \mathsf{min}\left(0, p\right)\right)\right)\right) \cdot p}{\mathsf{min}\left(0, p\right) - 1} \]

      10. sub-negate-rev N/A

          \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(\color{blue}{0}, p\right) - 1} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(\color{blue}{0}, p\right) - 1} \]

      12. lower-*.f64 26.5%

          \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1} \]

   8. Applied rewrites 26.5%

      \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(0, p\right) - 1\\ \mathbf{if}\;p \leq 600000000000:\\ \;\;\;\;\mathsf{min}\left(0, p\right) \cdot \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)} - -1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot p}{t\_0}\\ \end{array} \]

FPCore

(FPCore (p)
  :precision binary64
  (let* ((t_0 (- (fmin 0.0 p) 1.0)))
  (if (<= p 600000000000.0)
    (- (* (fmin 0.0 p) (/ (fmin 0.0 p) (- 1.0 (fmin 0.0 p)))) -1.0)
    (/ (* t_0 p) t_0))))

C

double code(double p) {
	double t_0 = fmin(0.0, p) - 1.0;
	double tmp;
	if (p <= 600000000000.0) {
		tmp = (fmin(0.0, p) * (fmin(0.0, p) / (1.0 - fmin(0.0, p)))) - -1.0;
	} else {
		tmp = (t_0 * p) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(p)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8) :: t_0
    real(8) :: tmp
    t_0 = fmin(0.0d0, p) - 1.0d0
    if (p <= 600000000000.0d0) then
        tmp = (fmin(0.0d0, p) * (fmin(0.0d0, p) / (1.0d0 - fmin(0.0d0, p)))) - (-1.0d0)
    else
        tmp = (t_0 * p) / t_0
    end if
    code = tmp
end function

Java

public static double code(double p) {
	double t_0 = fmin(0.0, p) - 1.0;
	double tmp;
	if (p <= 600000000000.0) {
		tmp = (fmin(0.0, p) * (fmin(0.0, p) / (1.0 - fmin(0.0, p)))) - -1.0;
	} else {
		tmp = (t_0 * p) / t_0;
	}
	return tmp;
}

Python

def code(p):
	t_0 = fmin(0.0, p) - 1.0
	tmp = 0
	if p <= 600000000000.0:
		tmp = (fmin(0.0, p) * (fmin(0.0, p) / (1.0 - fmin(0.0, p)))) - -1.0
	else:
		tmp = (t_0 * p) / t_0
	return tmp

Julia

function code(p)
	t_0 = Float64(fmin(0.0, p) - 1.0)
	tmp = 0.0
	if (p <= 600000000000.0)
		tmp = Float64(Float64(fmin(0.0, p) * Float64(fmin(0.0, p) / Float64(1.0 - fmin(0.0, p)))) - -1.0);
	else
		tmp = Float64(Float64(t_0 * p) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p)
	t_0 = min(0.0, p) - 1.0;
	tmp = 0.0;
	if (p <= 600000000000.0)
		tmp = (min(0.0, p) * (min(0.0, p) / (1.0 - min(0.0, p)))) - -1.0;
	else
		tmp = (t_0 * p) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_] := Block[{t$95$0 = N[(N[Min[0.0, p], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[p, 600000000000.0], N[(N[(N[Min[0.0, p], $MachinePrecision] * N[(N[Min[0.0, p], $MachinePrecision] / N[(1.0 - N[Min[0.0, p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision], N[(N[(t$95$0 * p), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if p < 6e11

   1. Initial program 76.6%

      \[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

      2. sub-flip N/A

         \[\leadsto \color{blue}{\left(p + 1\right) + \left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right) + \left(p + 1\right)} \]

      4. add-flip N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right) - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}\right)\right) - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}}\right)\right) - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      7. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{min}\left(p, 0\right) \cdot \mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{min}\left(p, 0\right) \cdot \mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      12. lift-fmin.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{min}\left(p, 0\right)} \cdot \mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      13. fmin-swap N/A

          \[\leadsto \frac{\color{blue}{\mathsf{min}\left(0, p\right)} \cdot \mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      14. lower-fmin.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{min}\left(0, p\right)} \cdot \mathsf{min}\left(p, 0\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      15. lift-fmin.f64 N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \color{blue}{\mathsf{min}\left(p, 0\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      16. fmin-swap N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      17. lower-fmin.f64 N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \color{blue}{\mathsf{min}\left(0, p\right)}}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      18. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{min}\left(p, 0\right) - 1\right)}\right)} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      19. sub-negate-rev N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{\color{blue}{1 - \mathsf{min}\left(p, 0\right)}} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      20. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{\color{blue}{1 - \mathsf{min}\left(p, 0\right)}} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      21. lift-fmin.f64 N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(p, 0\right)}} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      22. fmin-swap N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(0, p\right)}} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      23. lower-fmin.f64 N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \color{blue}{\mathsf{min}\left(0, p\right)}} - \left(\mathsf{neg}\left(\left(p + 1\right)\right)\right) \]

      24. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)} - \left(\mathsf{neg}\left(\color{blue}{\left(p + 1\right)}\right)\right) \]

      25. add-flip N/A

          \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)} - \left(\mathsf{neg}\left(\color{blue}{\left(p - \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

   3. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)} - \left(-1 - p\right)} \]

   4. Taylor expanded in p around 0

      \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)} - \color{blue}{-1} \]
   5. Step-by-step derivation

   6. Applied rewrites 51.9%

      \[\leadsto \frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)} - \color{blue}{-1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}} - -1 \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)}}{1 - \mathsf{min}\left(0, p\right)} - -1 \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\mathsf{min}\left(0, p\right) \cdot \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}} - -1 \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{min}\left(0, p\right) \cdot \color{blue}{\frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}} - -1 \]

      5. lower-*.f64 51.9%

         \[\leadsto \color{blue}{\mathsf{min}\left(0, p\right) \cdot \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}} - -1 \]

   8. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\mathsf{min}\left(0, p\right) \cdot \frac{\mathsf{min}\left(0, p\right)}{1 - \mathsf{min}\left(0, p\right)}} - -1 \]

   if 6e11 < p

   1. Initial program 76.6%

      \[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(p + 1\right) - \color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

      3. frac-2neg N/A

         \[\leadsto \left(p + 1\right) - \color{blue}{\frac{\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

      4. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(p + 1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right) - \left(\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

      5. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(p + 1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right) - \left(\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(p + 1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right) - \left(\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

   3. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(p - -1, 1 - \mathsf{min}\left(0, p\right), \mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)\right) \cdot \frac{-1}{\mathsf{min}\left(0, p\right) - 1}} \]

   4. Taylor expanded in p around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1} \]

      4. lower--.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, \color{blue}{p}\right) - 1} \]

      5. lower-fmin.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1} \]

      6. lower--.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - \color{blue}{1}} \]

      7. lower-fmin.f64 26.5%

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1} \]

   6. Applied rewrites 26.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]

      2. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot \left(p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)\right)}{\mathsf{min}\left(\color{blue}{0}, p\right) - 1} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 - \mathsf{min}\left(0, p\right)\right) \cdot p\right)}{\mathsf{min}\left(\color{blue}{0}, p\right) - 1} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - \mathsf{min}\left(0, p\right)\right)\right)\right) \cdot p}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - \mathsf{min}\left(0, p\right)\right)\right)\right) \cdot p}{\mathsf{min}\left(0, p\right) - 1} \]

      10. sub-negate-rev N/A

          \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(\color{blue}{0}, p\right) - 1} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(\color{blue}{0}, p\right) - 1} \]

      12. lower-*.f64 26.5%

          \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1} \]

   8. Applied rewrites 26.5%

      \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 26.5% accurate, 1.8× speedup

Math

\[\left(-p\right) \cdot \frac{\mathsf{min}\left(0, p\right) - 1}{1 - \mathsf{min}\left(0, p\right)} \]

FPCore

(FPCore (p)
  :precision binary64
  (* (- p) (/ (- (fmin 0.0 p) 1.0) (- 1.0 (fmin 0.0 p)))))

C

double code(double p) {
	return -p * ((fmin(0.0, p) - 1.0) / (1.0 - fmin(0.0, p)));
}

Fortran

real(8) function code(p)
use fmin_fmax_functions
    real(8), intent (in) :: p
    code = -p * ((fmin(0.0d0, p) - 1.0d0) / (1.0d0 - fmin(0.0d0, p)))
end function

Java

public static double code(double p) {
	return -p * ((fmin(0.0, p) - 1.0) / (1.0 - fmin(0.0, p)));
}

Python

def code(p):
	return -p * ((fmin(0.0, p) - 1.0) / (1.0 - fmin(0.0, p)))

Julia

function code(p)
	return Float64(Float64(-p) * Float64(Float64(fmin(0.0, p) - 1.0) / Float64(1.0 - fmin(0.0, p))))
end

MATLAB

function tmp = code(p)
	tmp = -p * ((min(0.0, p) - 1.0) / (1.0 - min(0.0, p)));
end

Wolfram

code[p_] := N[((-p) * N[(N[(N[Min[0.0, p], $MachinePrecision] - 1.0), $MachinePrecision] / N[(1.0 - N[Min[0.0, p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

   2. lift-/.f64 N/A

      \[\leadsto \left(p + 1\right) - \color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

   3. frac-2neg N/A

      \[\leadsto \left(p + 1\right) - \color{blue}{\frac{\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

   4. sub-to-fraction N/A

      \[\leadsto \color{blue}{\frac{\left(p + 1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right) - \left(\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

   5. mult-flip N/A

      \[\leadsto \color{blue}{\left(\left(p + 1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right) - \left(\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(p + 1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right) - \left(\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

3. Applied rewrites 76.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(p - -1, 1 - \mathsf{min}\left(0, p\right), \mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)\right) \cdot \frac{-1}{\mathsf{min}\left(0, p\right) - 1}} \]

4. Taylor expanded in p around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]

   2. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]

   3. lower-*.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1} \]

   4. lower--.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, \color{blue}{p}\right) - 1} \]

   5. lower-fmin.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1} \]

   6. lower--.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - \color{blue}{1}} \]

   7. lower-fmin.f64 26.5%

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1} \]

6. Applied rewrites 26.5%

   \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(p \cdot \frac{1 - \mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1}\right) \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \left(\mathsf{neg}\left(p\right)\right) \cdot \color{blue}{\frac{1 - \mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1}} \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(p\right)\right) \cdot \color{blue}{\frac{1 - \mathsf{min}\left(0, p\right)}{\mathsf{min}\left(0, p\right) - 1}} \]

   8. lower-neg.f64 N/A

      \[\leadsto \left(-p\right) \cdot \frac{\color{blue}{1 - \mathsf{min}\left(0, p\right)}}{\mathsf{min}\left(0, p\right) - 1} \]

   9. lift--.f64 N/A

      \[\leadsto \left(-p\right) \cdot \frac{1 - \mathsf{min}\left(0, p\right)}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1} \]

   10. sub-negate-rev N/A

       \[\leadsto \left(-p\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{min}\left(0, p\right) - 1\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1} \]

   11. lift--.f64 N/A

       \[\leadsto \left(-p\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{min}\left(0, p\right) - 1\right)\right)}{\mathsf{min}\left(\color{blue}{0}, p\right) - 1} \]

   12. lift--.f64 N/A

       \[\leadsto \left(-p\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{min}\left(0, p\right) - 1\right)\right)}{\mathsf{min}\left(0, p\right) - \color{blue}{1}} \]

   13. sub-negate-rev N/A

       \[\leadsto \left(-p\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{min}\left(0, p\right) - 1\right)\right)}{\mathsf{neg}\left(\left(1 - \mathsf{min}\left(0, p\right)\right)\right)} \]

   14. lift--.f64 N/A

       \[\leadsto \left(-p\right) \cdot \frac{\mathsf{neg}\left(\left(\mathsf{min}\left(0, p\right) - 1\right)\right)}{\mathsf{neg}\left(\left(1 - \mathsf{min}\left(0, p\right)\right)\right)} \]

   15. frac-2neg-rev N/A

       \[\leadsto \left(-p\right) \cdot \frac{\mathsf{min}\left(0, p\right) - 1}{\color{blue}{1 - \mathsf{min}\left(0, p\right)}} \]

   16. lower-/.f64 26.5%

       \[\leadsto \left(-p\right) \cdot \frac{\mathsf{min}\left(0, p\right) - 1}{\color{blue}{1 - \mathsf{min}\left(0, p\right)}} \]

8. Applied rewrites 26.5%

   \[\leadsto \left(-p\right) \cdot \color{blue}{\frac{\mathsf{min}\left(0, p\right) - 1}{1 - \mathsf{min}\left(0, p\right)}} \]
9. Add Preprocessing

Alternative 7: 26.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(0, p\right) - 1\\ \frac{t\_0 \cdot p}{t\_0} \end{array} \]

FPCore

(FPCore (p)
  :precision binary64
  (let* ((t_0 (- (fmin 0.0 p) 1.0))) (/ (* t_0 p) t_0)))

C

double code(double p) {
	double t_0 = fmin(0.0, p) - 1.0;
	return (t_0 * p) / t_0;
}

Fortran

real(8) function code(p)
use fmin_fmax_functions
    real(8), intent (in) :: p
    real(8) :: t_0
    t_0 = fmin(0.0d0, p) - 1.0d0
    code = (t_0 * p) / t_0
end function

Java

public static double code(double p) {
	double t_0 = fmin(0.0, p) - 1.0;
	return (t_0 * p) / t_0;
}

Python

def code(p):
	t_0 = fmin(0.0, p) - 1.0
	return (t_0 * p) / t_0

Julia

function code(p)
	t_0 = Float64(fmin(0.0, p) - 1.0)
	return Float64(Float64(t_0 * p) / t_0)
end

MATLAB

function tmp = code(p)
	t_0 = min(0.0, p) - 1.0;
	tmp = (t_0 * p) / t_0;
end

Wolfram

code[p_] := Block[{t$95$0 = N[(N[Min[0.0, p], $MachinePrecision] - 1.0), $MachinePrecision]}, N[(N[(t$95$0 * p), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 76.6%

   \[\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1} \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(p + 1\right) - \frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

   2. lift-/.f64 N/A

      \[\leadsto \left(p + 1\right) - \color{blue}{\frac{{\left(\mathsf{min}\left(p, 0\right)\right)}^{2}}{\mathsf{min}\left(p, 0\right) - 1}} \]

   3. frac-2neg N/A

      \[\leadsto \left(p + 1\right) - \color{blue}{\frac{\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

   4. sub-to-fraction N/A

      \[\leadsto \color{blue}{\frac{\left(p + 1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right) - \left(\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

   5. mult-flip N/A

      \[\leadsto \color{blue}{\left(\left(p + 1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right) - \left(\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(p + 1\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)\right) - \left(\mathsf{neg}\left({\left(\mathsf{min}\left(p, 0\right)\right)}^{2}\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left(\mathsf{min}\left(p, 0\right) - 1\right)\right)}} \]

3. Applied rewrites 76.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(p - -1, 1 - \mathsf{min}\left(0, p\right), \mathsf{min}\left(0, p\right) \cdot \mathsf{min}\left(0, p\right)\right) \cdot \frac{-1}{\mathsf{min}\left(0, p\right) - 1}} \]

4. Taylor expanded in p around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]

   2. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]

   3. lower-*.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1} \]

   4. lower--.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, \color{blue}{p}\right) - 1} \]

   5. lower-fmin.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1} \]

   6. lower--.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - \color{blue}{1}} \]

   7. lower-fmin.f64 26.5%

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1} \]

6. Applied rewrites 26.5%

   \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\mathsf{min}\left(0, p\right) - 1}} \]

   2. lift-/.f64 N/A

      \[\leadsto -1 \cdot \frac{p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{-1 \cdot \left(p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]

   4. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)\right)}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(p \cdot \left(1 - \mathsf{min}\left(0, p\right)\right)\right)}{\mathsf{min}\left(\color{blue}{0}, p\right) - 1} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(1 - \mathsf{min}\left(0, p\right)\right) \cdot p\right)}{\mathsf{min}\left(\color{blue}{0}, p\right) - 1} \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - \mathsf{min}\left(0, p\right)\right)\right)\right) \cdot p}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1} \]

   9. lift--.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\left(1 - \mathsf{min}\left(0, p\right)\right)\right)\right) \cdot p}{\mathsf{min}\left(0, p\right) - 1} \]

   10. sub-negate-rev N/A

       \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(\color{blue}{0}, p\right) - 1} \]

   11. lift--.f64 N/A

       \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\mathsf{min}\left(\color{blue}{0}, p\right) - 1} \]

   12. lower-*.f64 26.5%

       \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\color{blue}{\mathsf{min}\left(0, p\right)} - 1} \]

8. Applied rewrites 26.5%

   \[\leadsto \frac{\left(\mathsf{min}\left(0, p\right) - 1\right) \cdot p}{\color{blue}{\mathsf{min}\left(0, p\right) - 1}} \]
9. Add Preprocessing

Developer Target 1: 52.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;p \geq -1 \cdot 10^{-6} \land p \leq 0:\\ \;\;\;\;\left(p + 1\right) - \frac{p \cdot p}{p - 1}\\ \mathbf{else}:\\ \;\;\;\;p - \mathsf{fma}\left(\frac{p}{p - 1}, p, -1\right)\\ \end{array} \]

FPCore

(FPCore (p)
  :precision binary64
  (if (and (>= p -1e-6) (<= p 0.0))
  (- (+ p 1.0) (/ (* p p) (- p 1.0)))
  (- p (fma (/ p (- p 1.0)) p -1.0))))

C

double code(double p) {
	double tmp;
	if ((p >= -1e-6) && (p <= 0.0)) {
		tmp = (p + 1.0) - ((p * p) / (p - 1.0));
	} else {
		tmp = p - fma((p / (p - 1.0)), p, -1.0);
	}
	return tmp;
}

Julia

function code(p)
	tmp = 0.0
	if ((p >= -1e-6) && (p <= 0.0))
		tmp = Float64(Float64(p + 1.0) - Float64(Float64(p * p) / Float64(p - 1.0)));
	else
		tmp = Float64(p - fma(Float64(p / Float64(p - 1.0)), p, -1.0));
	end
	return tmp
end

Wolfram

code[p_] := If[And[GreaterEqual[p, -1e-6], LessEqual[p, 0.0]], N[(N[(p + 1.0), $MachinePrecision] - N[(N[(p * p), $MachinePrecision] / N[(p - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(p - N[(N[(p / N[(p - 1.0), $MachinePrecision]), $MachinePrecision] * p + -1.0), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025326 
(FPCore (p)
  :name "peicewise-defined"
  :precision binary64
  :pre (and (>= p -1.79e+308) (<= p 1.79e+308))

  :alt
  (if (and (>= p -1/1000000) (<= p 0)) (- (+ p 1) (/ (* p p) (- p 1))) (- p (fma (/ p (- p 1)) p -1)))

  (- (+ p 1.0) (/ (pow (fmin p 0.0) 2.0) (- (fmin p 0.0) 1.0))))