xlohi (overflows)

Percentage Accurate: 3.1% → 21.9%

Time: 9.5s

Alternatives: 5

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: 21.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in hi around inf 18.8%

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

   1. expm1-log1p-u 18.8%

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

   2. expm1-udef 18.8%

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

4. Applied egg-rr 18.8%

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

   1. flip3-- 18.8%

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

   2. div-inv 18.8%

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

   3. pow3 18.8%

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

   4. metadata-eval 18.8%

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

   5. sub-neg 18.8%

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

   6. pow3 18.8%

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

   7. log1p-udef 18.8%

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

   8. add-exp-log 18.8%

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

   9. +-commutative 18.8%

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

   10. metadata-eval 18.8%

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

6. Applied egg-rr 18.8%

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

7. Taylor expanded in hi around inf 21.8%

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

   1. +-commutative 21.8%

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

   2. associate--l+ 21.8%

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

   3. distribute-rgt1-in 21.8%

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

   4. metadata-eval 21.8%

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

9. Simplified 21.8%

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

10. Final simplification 21.8%

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

Alternative 2: 21.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(-0.5 \cdot \left(\frac{lo}{hi} \cdot \frac{lo}{hi}\right) - \frac{lo}{hi}\right) \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (expm1 (- (* -0.5 (* (/ lo hi) (/ lo hi))) (/ lo hi))))

C

double code(double lo, double hi, double x) {
	return expm1(((-0.5 * ((lo / hi) * (lo / hi))) - (lo / hi)));
}

Java

public static double code(double lo, double hi, double x) {
	return Math.expm1(((-0.5 * ((lo / hi) * (lo / hi))) - (lo / hi)));
}

Python

def code(lo, hi, x):
	return math.expm1(((-0.5 * ((lo / hi) * (lo / hi))) - (lo / hi)))

Julia

function code(lo, hi, x)
	return expm1(Float64(Float64(-0.5 * Float64(Float64(lo / hi) * Float64(lo / hi))) - Float64(lo / hi)))
end

Wolfram

code[lo_, hi_, x_] := N[(Exp[N[(N[(-0.5 * N[(N[(lo / hi), $MachinePrecision] * N[(lo / hi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(lo / hi), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in hi around inf 18.8%

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

   1. expm1-log1p-u 18.8%

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

   2. expm1-udef 18.8%

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

4. Applied egg-rr 18.8%

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

5. Taylor expanded in hi around inf 0.0%

   \[\leadsto e^{\color{blue}{\left(\frac{x}{hi} + -0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}}\right) - \frac{lo}{hi}}} - 1 \]
6. Step-by-step derivation

   1. associate--l+ 0.0%

      \[\leadsto e^{\color{blue}{\frac{x}{hi} + \left(-0.5 \cdot \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}} - \frac{lo}{hi}\right)}} - 1 \]

   2. fma-neg 0.0%

      \[\leadsto e^{\frac{x}{hi} + \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(x - lo\right)}^{2}}{{hi}^{2}}, -\frac{lo}{hi}\right)}} - 1 \]

   3. unpow2 0.0%

      \[\leadsto e^{\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(x - lo\right) \cdot \left(x - lo\right)}}{{hi}^{2}}, -\frac{lo}{hi}\right)} - 1 \]

   4. unpow2 0.0%

      \[\leadsto e^{\frac{x}{hi} + \mathsf{fma}\left(-0.5, \frac{\left(x - lo\right) \cdot \left(x - lo\right)}{\color{blue}{hi \cdot hi}}, -\frac{lo}{hi}\right)} - 1 \]

   5. times-frac 21.8%

      \[\leadsto e^{\frac{x}{hi} + \mathsf{fma}\left(-0.5, \color{blue}{\frac{x - lo}{hi} \cdot \frac{x - lo}{hi}}, -\frac{lo}{hi}\right)} - 1 \]

   6. unpow2 21.8%

      \[\leadsto e^{\frac{x}{hi} + \mathsf{fma}\left(-0.5, \color{blue}{{\left(\frac{x - lo}{hi}\right)}^{2}}, -\frac{lo}{hi}\right)} - 1 \]

   7. distribute-neg-frac 21.8%

      \[\leadsto e^{\frac{x}{hi} + \mathsf{fma}\left(-0.5, {\left(\frac{x - lo}{hi}\right)}^{2}, \color{blue}{\frac{-lo}{hi}}\right)} - 1 \]

7. Simplified 21.8%

   \[\leadsto e^{\color{blue}{\frac{x}{hi} + \mathsf{fma}\left(-0.5, {\left(\frac{x - lo}{hi}\right)}^{2}, \frac{-lo}{hi}\right)}} - 1 \]

8. Taylor expanded in x around 0 0.0%

   \[\leadsto \color{blue}{e^{-1 \cdot \frac{lo}{hi} + -0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}}} - 1} \]
9. Step-by-step derivation

   1. expm1-def 0.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(-1 \cdot \frac{lo}{hi} + -0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}}\right)} \]

   2. +-commutative 0.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}} + -1 \cdot \frac{lo}{hi}}\right) \]

   3. neg-mul-1 0.0%

      \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}} + \color{blue}{\left(-\frac{lo}{hi}\right)}\right) \]

   4. unsub-neg 0.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-0.5 \cdot \frac{{lo}^{2}}{{hi}^{2}} - \frac{lo}{hi}}\right) \]

   5. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \frac{\color{blue}{lo \cdot lo}}{{hi}^{2}} - \frac{lo}{hi}\right) \]

   6. unpow2 0.0%

      \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \frac{lo \cdot lo}{\color{blue}{hi \cdot hi}} - \frac{lo}{hi}\right) \]

   7. times-frac 21.8%

      \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \color{blue}{\left(\frac{lo}{hi} \cdot \frac{lo}{hi}\right)} - \frac{lo}{hi}\right) \]

10. Simplified 21.8%

    \[\leadsto \color{blue}{\mathsf{expm1}\left(-0.5 \cdot \left(\frac{lo}{hi} \cdot \frac{lo}{hi}\right) - \frac{lo}{hi}\right)} \]

11. Final simplification 21.8%

    \[\leadsto \mathsf{expm1}\left(-0.5 \cdot \left(\frac{lo}{hi} \cdot \frac{lo}{hi}\right) - \frac{lo}{hi}\right) \]

Alternative 3: 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. Taylor expanded in hi around inf 18.8%

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

3. Final simplification 18.8%

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

Alternative 4: 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(Float64(-lo) / hi)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[\frac{x - lo}{hi - lo} \]

2. Taylor expanded in lo around 0 18.8%

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

   1. mul-1-neg 18.8%

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

   2. unsub-neg 18.8%

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

   3. mul-1-neg 18.8%

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

   4. unsub-neg 18.8%

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

   5. unpow2 18.8%

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

4. Simplified 18.8%

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

5. Taylor expanded in x around 0 18.8%

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

   1. neg-mul-1 18.8%

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

   2. distribute-neg-frac 18.8%

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

7. Simplified 18.8%

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

8. Final simplification 18.8%

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

Alternative 5: 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. Taylor expanded in lo around inf 18.7%

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

3. Final simplification 18.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023258 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))