xlohi (overflows)

Percentage Accurate: 3.1% → 98.7%

Time: 10.2s

Alternatives: 7

Speedup: 18.0×

Specification

\[lo < -1 \cdot 10^{+308} \land hi > 10^{+308}\]

Math

\[\begin{array}{l} \\ \frac{x - lo}{hi - lo} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))

C

double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (x - lo) / (hi - lo)
end function

Java

public static double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

Python

def code(lo, hi, x):
	return (x - lo) / (hi - lo)

Julia

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (x - lo) / (hi - lo);
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]

Initial Program: 3.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - lo}{hi - lo} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (/ (- x lo) (- hi lo)))

C

double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (x - lo) / (hi - lo)
end function

Java

public static double code(double lo, double hi, double x) {
	return (x - lo) / (hi - lo);
}

Python

def code(lo, hi, x):
	return (x - lo) / (hi - lo)

Julia

function code(lo, hi, x)
	return Float64(Float64(x - lo) / Float64(hi - lo))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (x - lo) / (hi - lo);
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / N[(hi - lo), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{lo}\\ t_1 := t\_0 \cdot hi\\ {\left(\mathsf{fma}\left(t\_0, t\_0 - \frac{\frac{t\_1 - x}{lo} - -1}{lo} \cdot hi, {\left(\frac{t\_1}{lo}\right)}^{2}\right)\right)}^{-1} \cdot \left({\left(\frac{\left(\frac{-1}{lo} - \frac{hi}{lo \cdot lo}\right) \cdot x}{lo} \cdot hi\right)}^{3} + {t\_0}^{3}\right) \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x lo))) (t_1 (* t_0 hi)))
   (*
    (pow
     (fma
      t_0
      (- t_0 (* (/ (- (/ (- t_1 x) lo) -1.0) lo) hi))
      (pow (/ t_1 lo) 2.0))
     -1.0)
    (+
     (pow (* (/ (* (- (/ -1.0 lo) (/ hi (* lo lo))) x) lo) hi) 3.0)
     (pow t_0 3.0)))))

C

double code(double lo, double hi, double x) {
	double t_0 = 1.0 - (x / lo);
	double t_1 = t_0 * hi;
	return pow(fma(t_0, (t_0 - (((((t_1 - x) / lo) - -1.0) / lo) * hi)), pow((t_1 / lo), 2.0)), -1.0) * (pow((((((-1.0 / lo) - (hi / (lo * lo))) * x) / lo) * hi), 3.0) + pow(t_0, 3.0));
}

Julia

function code(lo, hi, x)
	t_0 = Float64(1.0 - Float64(x / lo))
	t_1 = Float64(t_0 * hi)
	return Float64((fma(t_0, Float64(t_0 - Float64(Float64(Float64(Float64(Float64(t_1 - x) / lo) - -1.0) / lo) * hi)), (Float64(t_1 / lo) ^ 2.0)) ^ -1.0) * Float64((Float64(Float64(Float64(Float64(Float64(-1.0 / lo) - Float64(hi / Float64(lo * lo))) * x) / lo) * hi) ^ 3.0) + (t_0 ^ 3.0)))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(1.0 - N[(x / lo), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * hi), $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[(t$95$0 - N[(N[(N[(N[(N[(t$95$1 - x), $MachinePrecision] / lo), $MachinePrecision] - -1.0), $MachinePrecision] / lo), $MachinePrecision] * hi), $MachinePrecision]), $MachinePrecision] + N[Power[N[(t$95$1 / lo), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[(N[Power[N[(N[(N[(N[(N[(-1.0 / lo), $MachinePrecision] - N[(hi / N[(lo * lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / lo), $MachinePrecision] * hi), $MachinePrecision], 3.0], $MachinePrecision] + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in hi around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{x - lo}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) + -1 \cdot \frac{x - lo}{lo}} \]

   2. mul-1-neg N/A

      \[\leadsto hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{x - lo}{lo}\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi} + \left(\mathsf{neg}\left(\frac{x - lo}{lo}\right)\right) \]

   4. div-sub N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{x}{lo} - \frac{lo}{lo}\right)}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{x}{lo} + \left(\mathsf{neg}\left(\frac{lo}{lo}\right)\right)\right)}\right)\right) \]

   6. *-inverses N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\left(\frac{x}{lo} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\left(\frac{x}{lo} + \color{blue}{-1}\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\color{blue}{\left(-1 + \frac{x}{lo}\right)}\right)\right) \]

   9. distribute-neg-in N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{x}{lo}\right)\right)\right)} \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\color{blue}{1} + \left(\mathsf{neg}\left(\frac{x}{lo}\right)\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(1 + \color{blue}{-1 \cdot \frac{x}{lo}}\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}, hi, 1 + -1 \cdot \frac{x}{lo}\right)} \]

5. Applied rewrites 18.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{lo} - \frac{\frac{x}{lo}}{lo}, hi, 1\right)}{lo} - \frac{\frac{x}{lo}}{lo}, hi, 1 - \frac{x}{lo}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 18.8%

   \[\leadsto \left({\left(1 - \frac{x}{lo}\right)}^{3} + {\left(\frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo} \cdot hi\right)}^{3}\right) \cdot \color{blue}{{\left(\mathsf{fma}\left(1 - \frac{x}{lo}, \left(1 - \frac{x}{lo}\right) - \frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo} \cdot hi, {\left(\frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo} \cdot hi\right)}^{2}\right)\right)}^{-1}} \]

8. Taylor expanded in hi around 0

   \[\leadsto \left({\left(1 - \frac{x}{lo}\right)}^{3} + {\left(\frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo} \cdot hi\right)}^{3}\right) \cdot {\left(\mathsf{fma}\left(1 - \frac{x}{lo}, \left(1 - \frac{x}{lo}\right) - \frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo} \cdot hi, {\left(\frac{hi \cdot \left(1 - \frac{x}{lo}\right)}{lo}\right)}^{2}\right)\right)}^{-1} \]
9. Step-by-step derivation

10. Applied rewrites 31.8%

    \[\leadsto \left({\left(1 - \frac{x}{lo}\right)}^{3} + {\left(\frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo} \cdot hi\right)}^{3}\right) \cdot {\left(\mathsf{fma}\left(1 - \frac{x}{lo}, \left(1 - \frac{x}{lo}\right) - \frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo} \cdot hi, {\left(\frac{\left(1 - \frac{x}{lo}\right) \cdot hi}{lo}\right)}^{2}\right)\right)}^{-1} \]

11. Taylor expanded in x around inf

    \[\leadsto \left({\left(1 - \frac{x}{lo}\right)}^{3} + {\left(\frac{x \cdot \left(-1 \cdot \frac{hi}{{lo}^{2}} - \frac{1}{lo}\right)}{lo} \cdot hi\right)}^{3}\right) \cdot {\left(\mathsf{fma}\left(1 - \frac{x}{lo}, \left(1 - \frac{x}{lo}\right) - \frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo} \cdot hi, {\left(\frac{\left(1 - \frac{x}{lo}\right) \cdot hi}{lo}\right)}^{2}\right)\right)}^{-1} \]
12. Step-by-step derivation

13. Applied rewrites 98.7%

    \[\leadsto \left({\left(1 - \frac{x}{lo}\right)}^{3} + {\left(\frac{\left(\frac{-hi}{lo \cdot lo} - \frac{1}{lo}\right) \cdot x}{lo} \cdot hi\right)}^{3}\right) \cdot {\left(\mathsf{fma}\left(1 - \frac{x}{lo}, \left(1 - \frac{x}{lo}\right) - \frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo} \cdot hi, {\left(\frac{\left(1 - \frac{x}{lo}\right) \cdot hi}{lo}\right)}^{2}\right)\right)}^{-1} \]

14. Final simplification 98.7%

    \[\leadsto {\left(\mathsf{fma}\left(1 - \frac{x}{lo}, \left(1 - \frac{x}{lo}\right) - \frac{\frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo} - -1}{lo} \cdot hi, {\left(\frac{\left(1 - \frac{x}{lo}\right) \cdot hi}{lo}\right)}^{2}\right)\right)}^{-1} \cdot \left({\left(\frac{\left(\frac{-1}{lo} - \frac{hi}{lo \cdot lo}\right) \cdot x}{lo} \cdot hi\right)}^{3} + {\left(1 - \frac{x}{lo}\right)}^{3}\right) \]
15. Add Preprocessing

Alternative 2: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\frac{hi + x}{lo}, -1, 1\right) - -2 \cdot \frac{x}{lo}} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (/ 1.0 (- (fma (/ (+ hi x) lo) -1.0 1.0) (* -2.0 (/ x lo)))))

C

double code(double lo, double hi, double x) {
	return 1.0 / (fma(((hi + x) / lo), -1.0, 1.0) - (-2.0 * (x / lo)));
}

Julia

function code(lo, hi, x)
	return Float64(1.0 / Float64(fma(Float64(Float64(hi + x) / lo), -1.0, 1.0) - Float64(-2.0 * Float64(x / lo))))
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 / N[(N[(N[(N[(hi + x), $MachinePrecision] / lo), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision] - N[(-2.0 * N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in hi around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{x - lo}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) + -1 \cdot \frac{x - lo}{lo}} \]

   2. mul-1-neg N/A

      \[\leadsto hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{x - lo}{lo}\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi} + \left(\mathsf{neg}\left(\frac{x - lo}{lo}\right)\right) \]

   4. div-sub N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{x}{lo} - \frac{lo}{lo}\right)}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{x}{lo} + \left(\mathsf{neg}\left(\frac{lo}{lo}\right)\right)\right)}\right)\right) \]

   6. *-inverses N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\left(\frac{x}{lo} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\left(\frac{x}{lo} + \color{blue}{-1}\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\color{blue}{\left(-1 + \frac{x}{lo}\right)}\right)\right) \]

   9. distribute-neg-in N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{x}{lo}\right)\right)\right)} \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\color{blue}{1} + \left(\mathsf{neg}\left(\frac{x}{lo}\right)\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(1 + \color{blue}{-1 \cdot \frac{x}{lo}}\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}, hi, 1 + -1 \cdot \frac{x}{lo}\right)} \]

5. Applied rewrites 18.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{lo} - \frac{\frac{x}{lo}}{lo}, hi, 1\right)}{lo} - \frac{\frac{x}{lo}}{lo}, hi, 1 - \frac{x}{lo}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 18.8%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo}, hi, -\left(1 - \frac{x}{lo}\right)\right)}{{\left(\frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo} \cdot hi\right)}^{2} - {\left(1 - \frac{x}{lo}\right)}^{2}}}} \]

8. Taylor expanded in lo around inf

   \[\leadsto \frac{1}{\frac{-1}{\color{blue}{{\left(\frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo} \cdot hi\right)}^{2}} - {\left(1 - \frac{x}{lo}\right)}^{2}}} \]
9. Step-by-step derivation

10. Applied rewrites 18.9%

    \[\leadsto \frac{1}{\frac{-1}{\color{blue}{{\left(\frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo} \cdot hi\right)}^{2}} - {\left(1 - \frac{x}{lo}\right)}^{2}}} \]

11. Taylor expanded in lo around inf

    \[\leadsto \frac{1}{\left(1 + -1 \cdot \frac{hi + x}{lo}\right) - \color{blue}{-2 \cdot \frac{x}{lo}}} \]
12. Step-by-step derivation

13. Applied rewrites 98.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{hi + x}{lo}, -1, 1\right) - \color{blue}{-2 \cdot \frac{x}{lo}}} \]
14. Add Preprocessing

Alternative 3: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(-hi\right) - x, -1, -2 \cdot x\right)}{lo}, -1, 1\right)} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (/ 1.0 (fma (/ (fma (- (- hi) x) -1.0 (* -2.0 x)) lo) -1.0 1.0)))

C

double code(double lo, double hi, double x) {
	return 1.0 / fma((fma((-hi - x), -1.0, (-2.0 * x)) / lo), -1.0, 1.0);
}

Julia

function code(lo, hi, x)
	return Float64(1.0 / fma(Float64(fma(Float64(Float64(-hi) - x), -1.0, Float64(-2.0 * x)) / lo), -1.0, 1.0))
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 / N[(N[(N[(N[((-hi) - x), $MachinePrecision] * -1.0 + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision] * -1.0 + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in hi around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{x - lo}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) + -1 \cdot \frac{x - lo}{lo}} \]

   2. mul-1-neg N/A

      \[\leadsto hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{x - lo}{lo}\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi} + \left(\mathsf{neg}\left(\frac{x - lo}{lo}\right)\right) \]

   4. div-sub N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{x}{lo} - \frac{lo}{lo}\right)}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{x}{lo} + \left(\mathsf{neg}\left(\frac{lo}{lo}\right)\right)\right)}\right)\right) \]

   6. *-inverses N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\left(\frac{x}{lo} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\left(\frac{x}{lo} + \color{blue}{-1}\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\color{blue}{\left(-1 + \frac{x}{lo}\right)}\right)\right) \]

   9. distribute-neg-in N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{x}{lo}\right)\right)\right)} \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\color{blue}{1} + \left(\mathsf{neg}\left(\frac{x}{lo}\right)\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(1 + \color{blue}{-1 \cdot \frac{x}{lo}}\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}, hi, 1 + -1 \cdot \frac{x}{lo}\right)} \]

5. Applied rewrites 18.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{lo} - \frac{\frac{x}{lo}}{lo}, hi, 1\right)}{lo} - \frac{\frac{x}{lo}}{lo}, hi, 1 - \frac{x}{lo}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 18.8%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo}, hi, -\left(1 - \frac{x}{lo}\right)\right)}{{\left(\frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo} \cdot hi\right)}^{2} - {\left(1 - \frac{x}{lo}\right)}^{2}}}} \]

8. Taylor expanded in lo around inf

   \[\leadsto \frac{1}{\frac{-1}{\color{blue}{{\left(\frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo} \cdot hi\right)}^{2}} - {\left(1 - \frac{x}{lo}\right)}^{2}}} \]
9. Step-by-step derivation

10. Applied rewrites 18.9%

    \[\leadsto \frac{1}{\frac{-1}{\color{blue}{{\left(\frac{1 + \frac{\left(1 - \frac{x}{lo}\right) \cdot hi - x}{lo}}{lo} \cdot hi\right)}^{2}} - {\left(1 - \frac{x}{lo}\right)}^{2}}} \]

11. Taylor expanded in lo around -inf

    \[\leadsto \frac{1}{1 + \color{blue}{-1 \cdot \frac{-1 \cdot \left(-1 \cdot hi + -1 \cdot x\right) - 2 \cdot x}{lo}}} \]
12. Step-by-step derivation

13. Applied rewrites 98.4%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-\left(hi + x\right), -1, -2 \cdot x\right)}{lo}, \color{blue}{-1}, 1\right)} \]

14. Final simplification 98.4%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(-hi\right) - x, -1, -2 \cdot x\right)}{lo}, -1, 1\right)} \]
15. Add Preprocessing

Alternative 4: 18.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\frac{hi}{lo} - -1}{lo}, hi, 1\right) \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (fma (/ (- (/ hi lo) -1.0) lo) hi 1.0))

C

double code(double lo, double hi, double x) {
	return fma((((hi / lo) - -1.0) / lo), hi, 1.0);
}

Julia

function code(lo, hi, x)
	return fma(Float64(Float64(Float64(hi / lo) - -1.0) / lo), hi, 1.0)
end

Wolfram

code[lo_, hi_, x_] := N[(N[(N[(N[(hi / lo), $MachinePrecision] - -1.0), $MachinePrecision] / lo), $MachinePrecision] * hi + 1.0), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in hi around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{x - lo}{lo} + hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) + -1 \cdot \frac{x - lo}{lo}} \]

   2. mul-1-neg N/A

      \[\leadsto hi \cdot \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{x - lo}{lo}\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi} + \left(\mathsf{neg}\left(\frac{x - lo}{lo}\right)\right) \]

   4. div-sub N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{x}{lo} - \frac{lo}{lo}\right)}\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{x}{lo} + \left(\mathsf{neg}\left(\frac{lo}{lo}\right)\right)\right)}\right)\right) \]

   6. *-inverses N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\left(\frac{x}{lo} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\left(\frac{x}{lo} + \color{blue}{-1}\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\mathsf{neg}\left(\color{blue}{\left(-1 + \frac{x}{lo}\right)}\right)\right) \]

   9. distribute-neg-in N/A

      \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(\frac{x}{lo}\right)\right)\right)} \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(\color{blue}{1} + \left(\mathsf{neg}\left(\frac{x}{lo}\right)\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}\right) \cdot hi + \left(1 + \color{blue}{-1 \cdot \frac{x}{lo}}\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{lo} + \frac{hi \cdot \left(\frac{1}{lo} - \frac{x}{{lo}^{2}}\right)}{lo}\right) - \frac{x}{{lo}^{2}}, hi, 1 + -1 \cdot \frac{x}{lo}\right)} \]

5. Applied rewrites 18.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{lo} - \frac{\frac{x}{lo}}{lo}, hi, 1\right)}{lo} - \frac{\frac{x}{lo}}{lo}, hi, 1 - \frac{x}{lo}\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto 1 + \color{blue}{hi \cdot \left(\frac{1}{lo} + \frac{hi}{{lo}^{2}}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 18.8%

   \[\leadsto \mathsf{fma}\left(\frac{\frac{hi}{lo} + 1}{lo}, \color{blue}{hi}, 1\right) \]

9. Final simplification 18.8%

   \[\leadsto \mathsf{fma}\left(\frac{\frac{hi}{lo} - -1}{lo}, hi, 1\right) \]
10. Add Preprocessing

Alternative 5: 18.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{x - lo}{hi} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (/ (- x lo) hi))

C

double code(double lo, double hi, double x) {
	return (x - lo) / hi;
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = (x - lo) / hi
end function

Java

public static double code(double lo, double hi, double x) {
	return (x - lo) / hi;
}

Python

def code(lo, hi, x):
	return (x - lo) / hi

Julia

function code(lo, hi, x)
	return Float64(Float64(x - lo) / hi)
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = (x - lo) / hi;
end

Wolfram

code[lo_, hi_, x_] := N[(N[(x - lo), $MachinePrecision] / hi), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in hi around inf

   \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]

   2. lower--.f64 18.8

      \[\leadsto \frac{\color{blue}{x - lo}}{hi} \]

5. Applied rewrites 18.8%

   \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]
6. Add Preprocessing

Alternative 6: 18.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{-lo}{hi} \end{array} \]

FPCore

(FPCore (lo hi x) :precision binary64 (/ (- lo) hi))

C

double code(double lo, double hi, double x) {
	return -lo / hi;
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = -lo / hi
end function

Java

public static double code(double lo, double hi, double x) {
	return -lo / hi;
}

Python

def code(lo, hi, x):
	return -lo / hi

Julia

function code(lo, hi, x)
	return Float64(Float64(-lo) / hi)
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = -lo / hi;
end

Wolfram

code[lo_, hi_, x_] := N[((-lo) / hi), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in hi around inf

   \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]

   2. lower--.f64 18.8

      \[\leadsto \frac{\color{blue}{x - lo}}{hi} \]

5. Applied rewrites 18.8%

   \[\leadsto \color{blue}{\frac{x - lo}{hi}} \]

6. Taylor expanded in lo around inf

   \[\leadsto \frac{-1 \cdot lo}{hi} \]
7. Step-by-step derivation

8. Applied rewrites 18.8%

   \[\leadsto \frac{-lo}{hi} \]
9. Add Preprocessing

Alternative 7: 18.7% accurate, 18.0× speedup

Math

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

FPCore

(FPCore (lo hi x) :precision binary64 1.0)

C

double code(double lo, double hi, double x) {
	return 1.0;
}

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0;
}

Python

def code(lo, hi, x):
	return 1.0

Julia

function code(lo, hi, x)
	return 1.0
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0;
end

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]
2. Add Preprocessing

3. Taylor expanded in lo around inf

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 18.7%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))