xlohi (overflows)

Percentage Accurate: 3.1% → 31.9%

Time: 22.3s

Alternatives: 11

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: 31.9% accurate, 0.0× speedup

Math

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

FPCore

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

C

double code(double lo, double hi, double x) {
	double t_0 = x / pow(lo, 2.0);
	return (-1.0 - pow((((x - hi) / lo) * (-1.0 - (hi / lo))), 3.0)) / (-1.0 - (((x * (1.0 + (x / lo))) / lo) - (hi * ((1.0 / lo) + (t_0 + ((x * ((1.0 / lo) - t_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
    real(8) :: t_0
    t_0 = x / (lo ** 2.0d0)
    code = ((-1.0d0) - ((((x - hi) / lo) * ((-1.0d0) - (hi / lo))) ** 3.0d0)) / ((-1.0d0) - (((x * (1.0d0 + (x / lo))) / lo) - (hi * ((1.0d0 / lo) + (t_0 + ((x * ((1.0d0 / lo) - t_0)) / lo))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(x / N[Power[lo, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(-1.0 - N[Power[N[(N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision] * N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(N[(N[(x * N[(1.0 + N[(x / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision] - N[(hi * N[(N[(1.0 / lo), $MachinePrecision] + N[(t$95$0 + N[(N[(x * N[(N[(1.0 / lo), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / lo), $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 18.9%

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

   1. flip3-+ 18.9%

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

   2. frac-2neg 18.9%

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

   3. metadata-eval 18.9%

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

   4. +-commutative 18.9%

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

   5. metadata-eval 18.9%

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

7. Applied egg-rr 18.9%

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

   1. distribute-neg-in 18.9%

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

   2. metadata-eval 18.9%

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

   3. unsub-neg 18.9%

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

   4. distribute-neg-in 18.9%

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

   5. metadata-eval 18.9%

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

   6. unsub-neg 18.9%

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

   7. unpow2 18.9%

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

   8. *-rgt-identity 18.9%

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

   9. distribute-lft-out-- 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in hi around 0 24.2%

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

11. Taylor expanded in hi around 0 32.4%

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

12. Final simplification 32.4%

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

Alternative 2: 31.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(N[(-1.0 - N[Power[N[(N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision] * N[(-1.0 - N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(-1.0 + 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 18.9%

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

   1. flip3-+ 18.9%

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

   2. frac-2neg 18.9%

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

   3. metadata-eval 18.9%

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

   4. +-commutative 18.9%

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

   5. metadata-eval 18.9%

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

7. Applied egg-rr 18.9%

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

   1. distribute-neg-in 18.9%

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

   2. metadata-eval 18.9%

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

   3. unsub-neg 18.9%

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

   4. distribute-neg-in 18.9%

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

   5. metadata-eval 18.9%

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

   6. unsub-neg 18.9%

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

   7. unpow2 18.9%

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

   8. *-rgt-identity 18.9%

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

   9. distribute-lft-out-- 18.9%

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

9. Simplified 18.9%

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

10. Taylor expanded in hi around 0 24.2%

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

11. Taylor expanded in lo around inf 32.4%

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

    1. mul-1-neg 32.4%

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

    2. distribute-neg-frac2 32.4%

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

13. Simplified 32.4%

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

14. Final simplification 32.4%

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

Alternative 3: 19.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(x * N[(N[(N[Power[N[(hi / lo), $MachinePrecision], 2.0], $MachinePrecision] / x), $MachinePrecision] + 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 18.9%

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

6. Taylor expanded in x around inf 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

8. Simplified 18.9%

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

9. Taylor expanded in hi around inf 0.0%

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

    1. associate-/r* 0.0%

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

    2. unpow2 0.0%

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

    3. unpow2 0.0%

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

    4. times-frac 19.5%

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

    5. unpow2 19.5%

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

11. Simplified 19.5%

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

12. Final simplification 19.5%

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

Alternative 4: 19.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(N[(hi - x), $MachinePrecision] / lo), $MachinePrecision] * N[Abs[N[(1.0 + N[(hi / lo), $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 18.9%

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

   1. add-sqr-sqrt 10.0%

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

   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. Final simplification 19.4%

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

Alternative 5: 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 18.9%

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

   1. add-sqr-sqrt 10.0%

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

   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 6: 18.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(x * N[(N[(N[(1.0 + N[(N[(hi * N[(1.0 + N[(hi / lo), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lo), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(-1.0 / 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 18.9%

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

6. Taylor expanded in x around inf 18.8%

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

   1. +-commutative 18.8%

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

   2. mul-1-neg 18.8%

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

   3. unsub-neg 18.8%

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

8. Simplified 18.9%

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

9. Taylor expanded in x around 0 18.9%

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

10. Taylor expanded in hi around 0 18.9%

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

11. Final simplification 18.9%

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

Alternative 7: 18.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := N[(1.0 + N[(N[(N[(x - hi), $MachinePrecision] / lo), $MachinePrecision] * N[(-1.0 - N[(hi / lo), $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 18.9%

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

6. Final simplification 18.9%

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

Alternative 8: 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 18.9%

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

6. Taylor expanded in x around 0 18.9%

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

   1. associate-/l* 18.9%

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

8. Simplified 18.9%

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

9. Final simplification 18.9%

   \[\leadsto 1 - hi \cdot \frac{-1 - \frac{hi}{lo}}{lo} \]
10. Add Preprocessing

Alternative 9: 18.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{x}{lo} \end{array} \]

FPCore

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

C

double code(double lo, double hi, double x) {
	return 1.0 - (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 - (x / lo)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around 0 18.7%

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

   1. div-sub 18.7%

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

   2. sub-neg 18.7%

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

   3. *-inverses 18.7%

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

   4. metadata-eval 18.7%

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

   5. distribute-lft-in 18.7%

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

   6. metadata-eval 18.7%

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

   7. +-commutative 18.7%

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

   8. mul-1-neg 18.7%

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

   9. unsub-neg 18.7%

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

5. Simplified 18.7%

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

6. Final simplification 18.7%

   \[\leadsto 1 - \frac{x}{lo} \]
7. Add Preprocessing

Alternative 10: 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 11: 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 2024067 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))