xlohi (overflows)

Percentage Accurate: 3.1% → 23.6%

Time: 20.4s

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: 23.6% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = hi * ((1.0d0 + (hi / lo)) / lo)
    code = (1.0d0 - (t_0 ** 3.0d0)) / (1.0d0 + ((hi * ((hi / lo) / lo)) + (t_0 ** 2.0d0)))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = hi * ((1.0 + (hi / lo)) / lo);
	return (1.0 - Math.pow(t_0, 3.0)) / (1.0 + ((hi * ((hi / lo) / lo)) + Math.pow(t_0, 2.0)));
}

Python

def code(lo, hi, x):
	t_0 = hi * ((1.0 + (hi / lo)) / lo)
	return (1.0 - math.pow(t_0, 3.0)) / (1.0 + ((hi * ((hi / lo) / lo)) + math.pow(t_0, 2.0)))

Julia

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

MATLAB

function tmp = code(lo, hi, x)
	t_0 = hi * ((1.0 + (hi / lo)) / lo);
	tmp = (1.0 - (t_0 ^ 3.0)) / (1.0 + ((hi * ((hi / lo) / lo)) + (t_0 ^ 2.0)));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(hi * N[(N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(hi * N[(N[(hi / lo), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $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. Taylor expanded in x around 0 18.8%

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

   1. associate-/l* 18.8%

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

8. Simplified 18.8%

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

   1. add-sqr-sqrt 18.8%

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

   2. sqrt-unprod 1.9%

      \[\leadsto 1 + \color{blue}{\sqrt{hi \cdot hi}} \cdot \frac{1 + \frac{hi}{lo}}{lo} \]

   3. sqr-neg 1.9%

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

   4. sqrt-unprod 0.0%

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

   5. add-sqr-sqrt 18.8%

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

   6. cancel-sign-sub-inv 18.8%

      \[\leadsto \color{blue}{1 - hi \cdot \frac{1 + \frac{hi}{lo}}{lo}} \]

   7. flip3-- 18.8%

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

   8. metadata-eval 18.8%

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

   9. metadata-eval 18.8%

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

   10. *-un-lft-identity 18.8%

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

   11. pow2 18.8%

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

10. Applied egg-rr 18.8%

    \[\leadsto \color{blue}{\frac{1 - {\left(hi \cdot \frac{1 + \frac{hi}{lo}}{lo}\right)}^{3}}{1 + \left({\left(hi \cdot \frac{1 + \frac{hi}{lo}}{lo}\right)}^{2} + hi \cdot \frac{1 + \frac{hi}{lo}}{lo}\right)}} \]
11. Step-by-step derivation

    1. +-commutative 18.8%

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

    2. +-commutative 18.8%

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

    3. +-commutative 18.8%

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

    4. +-commutative 18.8%

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

12. Simplified 18.8%

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

13. Taylor expanded in hi around inf 23.5%

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

14. Final simplification 23.5%

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

Alternative 2: 22.2% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lo, hi, x)
    real(8), intent (in) :: lo
    real(8), intent (in) :: hi
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = hi * ((1.0d0 + (hi / lo)) / lo)
    code = (1.0d0 - (t_0 ** 3.0d0)) / (1.0d0 + (t_0 + ((hi * (1.0d0 / lo)) ** 2.0d0)))
end function

Java

public static double code(double lo, double hi, double x) {
	double t_0 = hi * ((1.0 + (hi / lo)) / lo);
	return (1.0 - Math.pow(t_0, 3.0)) / (1.0 + (t_0 + Math.pow((hi * (1.0 / lo)), 2.0)));
}

Python

def code(lo, hi, x):
	t_0 = hi * ((1.0 + (hi / lo)) / lo)
	return (1.0 - math.pow(t_0, 3.0)) / (1.0 + (t_0 + math.pow((hi * (1.0 / lo)), 2.0)))

Julia

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

MATLAB

function tmp = code(lo, hi, x)
	t_0 = hi * ((1.0 + (hi / lo)) / lo);
	tmp = (1.0 - (t_0 ^ 3.0)) / (1.0 + (t_0 + ((hi * (1.0 / lo)) ^ 2.0)));
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(hi * N[(N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$0 + N[Power[N[(hi * N[(1.0 / lo), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $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. Taylor expanded in x around 0 18.8%

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

   1. associate-/l* 18.8%

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

8. Simplified 18.8%

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

   1. add-sqr-sqrt 18.8%

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

   2. sqrt-unprod 1.9%

      \[\leadsto 1 + \color{blue}{\sqrt{hi \cdot hi}} \cdot \frac{1 + \frac{hi}{lo}}{lo} \]

   3. sqr-neg 1.9%

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

   4. sqrt-unprod 0.0%

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

   5. add-sqr-sqrt 18.8%

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

   6. cancel-sign-sub-inv 18.8%

      \[\leadsto \color{blue}{1 - hi \cdot \frac{1 + \frac{hi}{lo}}{lo}} \]

   7. flip3-- 18.8%

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

   8. metadata-eval 18.8%

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

   9. metadata-eval 18.8%

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

   10. *-un-lft-identity 18.8%

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

   11. pow2 18.8%

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

10. Applied egg-rr 18.8%

    \[\leadsto \color{blue}{\frac{1 - {\left(hi \cdot \frac{1 + \frac{hi}{lo}}{lo}\right)}^{3}}{1 + \left({\left(hi \cdot \frac{1 + \frac{hi}{lo}}{lo}\right)}^{2} + hi \cdot \frac{1 + \frac{hi}{lo}}{lo}\right)}} \]
11. Step-by-step derivation

    1. +-commutative 18.8%

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

    2. +-commutative 18.8%

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

    3. +-commutative 18.8%

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

    4. +-commutative 18.8%

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

12. Simplified 18.8%

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

13. Taylor expanded in hi around 0 22.1%

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

14. Final simplification 22.1%

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

Alternative 3: 19.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ 1 + \left(hi - x\right) \cdot \frac{1 + {\left(\frac{hi}{lo}\right)}^{3}}{lo \cdot \left(1 + \left(\frac{hi}{lo} + {\left(\frac{hi}{lo}\right)}^{2}\right)\right)} \end{array} \]

FPCore

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

C

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

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) * ((1.0d0 + ((hi / lo) ** 3.0d0)) / (lo * (1.0d0 + ((hi / lo) + ((hi / lo) ** 2.0d0))))))
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0 + ((hi - x) * ((1.0 + Math.pow((hi / lo), 3.0)) / (lo * (1.0 + ((hi / lo) + Math.pow((hi / lo), 2.0))))));
}

Python

def code(lo, hi, x):
	return 1.0 + ((hi - x) * ((1.0 + math.pow((hi / lo), 3.0)) / (lo * (1.0 + ((hi / lo) + math.pow((hi / lo), 2.0))))))

Julia

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

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0 + ((hi - x) * ((1.0 + ((hi / lo) ^ 3.0)) / (lo * (1.0 + ((hi / lo) + ((hi / lo) ^ 2.0))))));
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(hi - x), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(hi / lo), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(lo * N[(1.0 + N[(N[(hi / lo), $MachinePrecision] + N[Power[N[(hi / lo), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. log1p-expm1-u 18.8%

      \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)\right)} \]

   2. log1p-undefine 18.8%

      \[\leadsto 1 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)\right)} \]

7. Applied egg-rr 18.8%

   \[\leadsto 1 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)\right)} \]
8. Step-by-step derivation

   1. log1p-define 18.8%

      \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\frac{hi}{lo} + 1\right) \cdot \frac{hi - x}{lo}\right)\right)} \]

   2. log1p-expm1-u 18.8%

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

   3. +-commutative 18.8%

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

   4. flip3-+ 18.8%

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

   5. frac-times 15.8%

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

   6. metadata-eval 15.8%

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

9. Applied egg-rr 16.9%

   \[\leadsto 1 + \color{blue}{\frac{\left(1 + {\left(\frac{hi}{lo}\right)}^{3}\right) \cdot \left(hi - x\right)}{\left(1 + \left({\left(\frac{hi}{lo}\right)}^{2} + \frac{hi}{lo}\right)\right) \cdot lo}} \]
10. Step-by-step derivation

    1. +-commutative 16.9%

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

    2. *-commutative 16.9%

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

    3. *-commutative 16.9%

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

    4. associate-+r+ 16.9%

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

    5. +-commutative 16.9%

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

    6. associate-+r+ 16.9%

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

    7. associate-/l* 19.5%

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

    8. +-commutative 19.5%

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

    9. associate-+r+ 19.5%

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

    10. +-commutative 19.5%

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

    11. associate-+r+ 19.5%

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

    12. +-commutative 19.5%

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

11. Simplified 19.5%

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

12. Final simplification 19.5%

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

Alternative 4: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return 1.0 + (fabs((2.0 + ((hi / lo) + -1.0))) * ((hi - x) / 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 + (abs((2.0d0 + ((hi / lo) + (-1.0d0)))) * ((hi - x) / lo))
end function

Java

public static double code(double lo, double hi, double x) {
	return 1.0 + (Math.abs((2.0 + ((hi / lo) + -1.0))) * ((hi - x) / lo));
}

Python

def code(lo, hi, x):
	return 1.0 + (math.fabs((2.0 + ((hi / lo) + -1.0))) * ((hi - x) / lo))

Julia

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

MATLAB

function tmp = code(lo, hi, x)
	tmp = 1.0 + (abs((2.0 + ((hi / lo) + -1.0))) * ((hi - x) / lo));
end

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[Abs[N[(2.0 + N[(N[(hi / lo), $MachinePrecision] + -1.0), $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. expm1-log1p-u 18.8%

      \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{hi}{lo} + 1\right)\right)} \cdot \frac{hi - x}{lo} \]

   2. expm1-undefine 18.8%

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

7. Applied egg-rr 18.8%

   \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{hi}{lo} + 1\right)} - 1\right)} \cdot \frac{hi - x}{lo} \]
8. Step-by-step derivation

   1. sub-neg 18.8%

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

   2. log1p-undefine 18.8%

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

   3. rem-exp-log 18.8%

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

   4. +-commutative 18.8%

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

   5. associate-+r+ 18.8%

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

   6. metadata-eval 18.8%

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

   7. metadata-eval 18.8%

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

9. Simplified 18.8%

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

    1. add-sqr-sqrt 9.8%

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

    2. sqrt-unprod 19.3%

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

    3. pow2 19.3%

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

    4. associate-+l+ 19.3%

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

11. Applied egg-rr 19.3%

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

    1. unpow2 19.3%

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

    2. rem-sqrt-square 19.3%

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

    3. +-commutative 19.3%

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

13. Simplified 19.3%

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

14. Final simplification 19.3%

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

Alternative 5: 19.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	return fabs(((hi - x) / 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 = abs(((hi - x) / lo))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[Abs[N[(N[(hi - x), $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 9.3%

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

   1. associate-+r- 9.3%

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

   2. distribute-lft-out-- 9.3%

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

   3. div-sub 9.3%

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

   4. mul-1-neg 9.3%

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

   5. unsub-neg 9.3%

      \[\leadsto \color{blue}{1 - \frac{x - hi}{lo}} \]

5. Simplified 9.3%

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

   1. add-sqr-sqrt 8.5%

      \[\leadsto \color{blue}{\sqrt{1 - \frac{x - hi}{lo}} \cdot \sqrt{1 - \frac{x - hi}{lo}}} \]

   2. sqrt-unprod 17.8%

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

   3. pow2 17.8%

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

7. Applied egg-rr 17.8%

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

   1. unpow2 17.8%

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

   2. rem-sqrt-square 17.8%

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

   3. div-sub 17.8%

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

   4. associate-+l- 17.8%

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

   5. sub-neg 17.8%

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

   6. associate-+r+ 17.8%

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

   7. +-commutative 17.8%

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

   8. sub-neg 17.8%

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

   9. div-sub 17.8%

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

9. Simplified 17.8%

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

10. Taylor expanded in lo around 0 19.3%

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

11. Final simplification 19.3%

    \[\leadsto \left|\frac{hi - x}{lo}\right| \]
12. Add Preprocessing

Alternative 6: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(hi * N[(N[(1.0 + N[(hi / lo), $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. Taylor expanded in x around 0 18.8%

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

   1. associate-/l* 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

   \[\leadsto 1 + hi \cdot \frac{1 + \frac{hi}{lo}}{lo} \]
10. 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 hi around inf 18.8%

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

4. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac2 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

   \[\leadsto \frac{lo}{-hi} \]
8. 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 18.7%

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

4. Final simplification 18.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024057 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))