xlohi (overflows)

Percentage Accurate: 3.1% → 20.6%

Time: 18.4s

Alternatives: 8

Speedup: 7.0×

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: 20.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ 1 + e^{3 \cdot \mathsf{log1p}\left(0.3333333333333333 \cdot \frac{hi}{lo}\right)} \cdot \frac{hi - x}{lo} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+
  1.0
  (* (exp (* 3.0 (log1p (* 0.3333333333333333 (/ hi lo))))) (/ (- hi x) lo))))

C

double code(double lo, double hi, double x) {
	return 1.0 + (exp((3.0 * log1p((0.3333333333333333 * (hi / lo))))) * ((hi - x) / lo));
}

Java

public static double code(double lo, double hi, double x) {
	return 1.0 + (Math.exp((3.0 * Math.log1p((0.3333333333333333 * (hi / lo))))) * ((hi - x) / lo));
}

Python

def code(lo, hi, x):
	return 1.0 + (math.exp((3.0 * math.log1p((0.3333333333333333 * (hi / lo))))) * ((hi - x) / lo))

Julia

function code(lo, hi, x)
	return Float64(1.0 + Float64(exp(Float64(3.0 * log1p(Float64(0.3333333333333333 * Float64(hi / lo))))) * Float64(Float64(hi - x) / lo)))
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[Exp[N[(3.0 * N[Log[1 + N[(0.3333333333333333 * N[(hi / lo), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.8%

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

   1. add-cube-cbrt 18.8%

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt[3]{\frac{hi}{lo} + 1} \cdot \sqrt[3]{\frac{hi}{lo} + 1}\right) \cdot \sqrt[3]{\frac{hi}{lo} + 1}\right)} \cdot \frac{hi - x}{lo} \]

   2. pow3 18.8%

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

   3. +-commutative 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in hi around 0 20.6%

   \[\leadsto 1 + {\color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{hi}{lo}\right)}}^{3} \cdot \frac{hi - x}{lo} \]
9. Step-by-step derivation

   1. add-exp-log 20.6%

      \[\leadsto 1 + \color{blue}{e^{\log \left({\left(1 + 0.3333333333333333 \cdot \frac{hi}{lo}\right)}^{3}\right)}} \cdot \frac{hi - x}{lo} \]

   2. log-pow 20.6%

      \[\leadsto 1 + e^{\color{blue}{3 \cdot \log \left(1 + 0.3333333333333333 \cdot \frac{hi}{lo}\right)}} \cdot \frac{hi - x}{lo} \]

   3. log1p-define 20.6%

      \[\leadsto 1 + e^{3 \cdot \color{blue}{\mathsf{log1p}\left(0.3333333333333333 \cdot \frac{hi}{lo}\right)}} \cdot \frac{hi - x}{lo} \]

10. Applied egg-rr 20.6%

    \[\leadsto 1 + \color{blue}{e^{3 \cdot \mathsf{log1p}\left(0.3333333333333333 \cdot \frac{hi}{lo}\right)}} \cdot \frac{hi - x}{lo} \]

11. Final simplification 20.6%

    \[\leadsto 1 + e^{3 \cdot \mathsf{log1p}\left(0.3333333333333333 \cdot \frac{hi}{lo}\right)} \cdot \frac{hi - x}{lo} \]
12. Add Preprocessing

Alternative 2: 20.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{hi - x}{lo} \cdot {\left(1 + 0.3333333333333333 \cdot \frac{hi}{lo}\right)}^{3} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+
  1.0
  (* (/ (- hi x) lo) (pow (+ 1.0 (* 0.3333333333333333 (/ hi lo))) 3.0))))

C

double code(double lo, double hi, double x) {
	return 1.0 + (((hi - x) / lo) * pow((1.0 + (0.3333333333333333 * (hi / lo))), 3.0));
}

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 + (((hi - x) / lo) * ((1.0d0 + (0.3333333333333333d0 * (hi / lo))) ** 3.0d0))
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0 + (((hi - x) / lo) * Math.pow((1.0 + (0.3333333333333333 * (hi / lo))), 3.0));
}

Python

def code(lo, hi, x):
	return 1.0 + (((hi - x) / lo) * math.pow((1.0 + (0.3333333333333333 * (hi / lo))), 3.0))

Julia

function code(lo, hi, x)
	return Float64(1.0 + Float64(Float64(Float64(hi - x) / lo) * (Float64(1.0 + Float64(0.3333333333333333 * Float64(hi / lo))) ^ 3.0)))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0 + (((hi - x) / lo) * ((1.0 + (0.3333333333333333 * (hi / lo))) ^ 3.0));
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision] * N[Power[N[(1.0 + N[(0.3333333333333333 * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.8%

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

   1. add-cube-cbrt 18.8%

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt[3]{\frac{hi}{lo} + 1} \cdot \sqrt[3]{\frac{hi}{lo} + 1}\right) \cdot \sqrt[3]{\frac{hi}{lo} + 1}\right)} \cdot \frac{hi - x}{lo} \]

   2. pow3 18.8%

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

   3. +-commutative 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in hi around 0 20.6%

   \[\leadsto 1 + {\color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{hi}{lo}\right)}}^{3} \cdot \frac{hi - x}{lo} \]

9. Final simplification 20.6%

   \[\leadsto 1 + \frac{hi - x}{lo} \cdot {\left(1 + 0.3333333333333333 \cdot \frac{hi}{lo}\right)}^{3} \]
10. Add Preprocessing

Alternative 3: 20.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{hi \cdot {\left(1 + 0.3333333333333333 \cdot \frac{hi}{lo}\right)}^{3}}{lo} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+ 1.0 (/ (* hi (pow (+ 1.0 (* 0.3333333333333333 (/ hi lo))) 3.0)) lo)))

C

double code(double lo, double hi, double x) {
	return 1.0 + ((hi * pow((1.0 + (0.3333333333333333 * (hi / lo))), 3.0)) / 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 = 1.0d0 + ((hi * ((1.0d0 + (0.3333333333333333d0 * (hi / lo))) ** 3.0d0)) / lo)
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0 + ((hi * Math.pow((1.0 + (0.3333333333333333 * (hi / lo))), 3.0)) / lo);
}

Python

def code(lo, hi, x):
	return 1.0 + ((hi * math.pow((1.0 + (0.3333333333333333 * (hi / lo))), 3.0)) / lo)

Julia

function code(lo, hi, x)
	return Float64(1.0 + Float64(Float64(hi * (Float64(1.0 + Float64(0.3333333333333333 * Float64(hi / lo))) ^ 3.0)) / lo))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0 + ((hi * ((1.0 + (0.3333333333333333 * (hi / lo))) ^ 3.0)) / lo);
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(hi * N[Power[N[(1.0 + N[(0.3333333333333333 * N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.8%

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

   1. add-cube-cbrt 18.8%

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt[3]{\frac{hi}{lo} + 1} \cdot \sqrt[3]{\frac{hi}{lo} + 1}\right) \cdot \sqrt[3]{\frac{hi}{lo} + 1}\right)} \cdot \frac{hi - x}{lo} \]

   2. pow3 18.8%

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

   3. +-commutative 18.8%

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

7. Applied egg-rr 18.8%

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

8. Taylor expanded in hi around 0 20.6%

   \[\leadsto 1 + {\color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{hi}{lo}\right)}}^{3} \cdot \frac{hi - x}{lo} \]

9. Taylor expanded in x around 0 20.6%

   \[\leadsto 1 + \color{blue}{\frac{hi \cdot {\left(1 + 0.3333333333333333 \cdot \frac{hi}{lo}\right)}^{3}}{lo}} \]

10. Final simplification 20.6%

    \[\leadsto 1 + \frac{hi \cdot {\left(1 + 0.3333333333333333 \cdot \frac{hi}{lo}\right)}^{3}}{lo} \]
11. Add Preprocessing

Alternative 4: 19.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 1 + hi \cdot \frac{\left|1 + \frac{hi}{lo}\right|}{lo} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+ 1.0 (* hi (/ (fabs (+ 1.0 (/ hi lo))) lo))))

C

double code(double lo, double hi, double x) {
	return 1.0 + (hi * (fabs((1.0 + (hi / lo))) / 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 = 1.0d0 + (hi * (abs((1.0d0 + (hi / lo))) / lo))
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0 + (hi * (Math.abs((1.0 + (hi / lo))) / lo));
}

Python

def code(lo, hi, x):
	return 1.0 + (hi * (math.fabs((1.0 + (hi / lo))) / lo))

Julia

function code(lo, hi, x)
	return Float64(1.0 + Float64(hi * Float64(abs(Float64(1.0 + Float64(hi / lo))) / lo)))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0 + (hi * (abs((1.0 + (hi / lo))) / lo));
end

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.8%

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

   1. add-sqr-sqrt 8.9%

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

   2. sqrt-unprod 19.4%

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

   3. pow2 19.4%

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

   4. +-commutative 19.4%

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

7. Applied egg-rr 19.4%

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

   1. unpow2 19.4%

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

   2. rem-sqrt-square 19.4%

      \[\leadsto 1 + \color{blue}{\left|1 + \frac{hi}{lo}\right|} \cdot \frac{hi - x}{lo} \]

9. Simplified 19.4%

   \[\leadsto 1 + \color{blue}{\left|1 + \frac{hi}{lo}\right|} \cdot \frac{hi - x}{lo} \]

10. Taylor expanded in hi around inf 19.4%

    \[\leadsto 1 + \color{blue}{\frac{hi \cdot \left|1 + \frac{hi}{lo}\right|}{lo}} \]
11. Step-by-step derivation

    1. associate-/l* 19.4%

       \[\leadsto 1 + \color{blue}{hi \cdot \frac{\left|1 + \frac{hi}{lo}\right|}{lo}} \]

12. Simplified 19.4%

    \[\leadsto 1 + \color{blue}{hi \cdot \frac{\left|1 + \frac{hi}{lo}\right|}{lo}} \]

13. Final simplification 19.4%

    \[\leadsto 1 + hi \cdot \frac{\left|1 + \frac{hi}{lo}\right|}{lo} \]
14. Add Preprocessing

Alternative 5: 19.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{hi \cdot \left|1 + \frac{hi}{lo}\right|}{lo} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (+ 1.0 (/ (* hi (fabs (+ 1.0 (/ hi lo)))) lo)))

C

double code(double lo, double hi, double x) {
	return 1.0 + ((hi * fabs((1.0 + (hi / lo)))) / 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 = 1.0d0 + ((hi * abs((1.0d0 + (hi / lo)))) / lo)
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0 + ((hi * Math.abs((1.0 + (hi / lo)))) / lo);
}

Python

def code(lo, hi, x):
	return 1.0 + ((hi * math.fabs((1.0 + (hi / lo)))) / lo)

Julia

function code(lo, hi, x)
	return Float64(1.0 + Float64(Float64(hi * abs(Float64(1.0 + Float64(hi / lo)))) / lo))
end

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0 + ((hi * abs((1.0 + (hi / lo)))) / lo);
end

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf 0.0%

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

5. Simplified 18.8%

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

   1. add-sqr-sqrt 8.9%

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

   2. sqrt-unprod 19.4%

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

   3. pow2 19.4%

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

   4. +-commutative 19.4%

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

7. Applied egg-rr 19.4%

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

   1. unpow2 19.4%

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

   2. rem-sqrt-square 19.4%

      \[\leadsto 1 + \color{blue}{\left|1 + \frac{hi}{lo}\right|} \cdot \frac{hi - x}{lo} \]

9. Simplified 19.4%

   \[\leadsto 1 + \color{blue}{\left|1 + \frac{hi}{lo}\right|} \cdot \frac{hi - x}{lo} \]

10. Taylor expanded in hi around inf 19.4%

    \[\leadsto 1 + \color{blue}{\frac{hi \cdot \left|1 + \frac{hi}{lo}\right|}{lo}} \]

11. Final simplification 19.4%

    \[\leadsto 1 + \frac{hi \cdot \left|1 + \frac{hi}{lo}\right|}{lo} \]
12. Add Preprocessing

Alternative 6: 18.8% accurate, 1.4× 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 18.8%

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

4. Final simplification 18.8%

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

Alternative 7: 18.8% accurate, 1.8× 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(lo / Float64(-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 lo around 0 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

   4. +-commutative 18.8%

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

   5. mul-1-neg 18.8%

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

   6. unsub-neg 18.8%

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

5. Simplified 18.8%

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

6. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

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

Alternative 8: 18.7% accurate, 7.0× 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 18.6%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 18.6%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))