xlohi (overflows)

Percentage Accurate: 3.1% → 26.9%

Time: 17.3s

Alternatives: 8

Speedup: 18.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 26.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{lo} + -1\\ t_1 := 1 - \frac{x}{lo}\\ t_2 := \frac{t\_1}{lo}\\ t_3 := \left(\frac{hi}{lo} + 1\right) \cdot t\_2\\ \left(\mathsf{fma}\left(hi, t\_3, t\_1\right) \cdot \mathsf{fma}\left(hi, t\_3, t\_0\right)\right) \cdot \frac{1}{\mathsf{fma}\left(hi, t\_2 \cdot 1, t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (lo hi x)
 :precision binary64
 (let* ((t_0 (+ (/ x lo) -1.0))
        (t_1 (- 1.0 (/ x lo)))
        (t_2 (/ t_1 lo))
        (t_3 (* (+ (/ hi lo) 1.0) t_2)))
   (* (* (fma hi t_3 t_1) (fma hi t_3 t_0)) (/ 1.0 (fma hi (* t_2 1.0) t_0)))))

C

double code(double lo, double hi, double x) {
	double t_0 = (x / lo) + -1.0;
	double t_1 = 1.0 - (x / lo);
	double t_2 = t_1 / lo;
	double t_3 = ((hi / lo) + 1.0) * t_2;
	return (fma(hi, t_3, t_1) * fma(hi, t_3, t_0)) * (1.0 / fma(hi, (t_2 * 1.0), t_0));
}

Julia

function code(lo, hi, x)
	t_0 = Float64(Float64(x / lo) + -1.0)
	t_1 = Float64(1.0 - Float64(x / lo))
	t_2 = Float64(t_1 / lo)
	t_3 = Float64(Float64(Float64(hi / lo) + 1.0) * t_2)
	return Float64(Float64(fma(hi, t_3, t_1) * fma(hi, t_3, t_0)) * Float64(1.0 / fma(hi, Float64(t_2 * 1.0), t_0)))
end

Wolfram

code[lo_, hi_, x_] := Block[{t$95$0 = N[(N[(x / lo), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(x / lo), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / lo), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(hi / lo), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision]}, N[(N[(N[(hi * t$95$3 + t$95$1), $MachinePrecision] * N[(hi * t$95$3 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(hi * N[(t$95$2 * 1.0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 18.8

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in hi around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. *-inverses N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-in N/A

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

   8. mul-1-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

8. Applied rewrites 18.8%

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

10. Applied rewrites 18.8%

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

11. Taylor expanded in hi around 0

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

13. Applied rewrites 26.7%

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

14. Final simplification 26.7%

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

Alternative 2: 26.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 18.8

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in hi around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. *-inverses N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-in N/A

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

   8. mul-1-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

8. Applied rewrites 18.8%

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

10. Applied rewrites 18.8%

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

11. Taylor expanded in lo around -inf

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

13. Applied rewrites 26.7%

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

14. Final simplification 26.7%

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

Alternative 3: 19.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 18.8

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in hi around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. *-inverses N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-in N/A

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

   8. mul-1-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

8. Applied rewrites 18.8%

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

10. Applied rewrites 18.8%

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

11. Taylor expanded in lo around inf

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

13. Applied rewrites 18.9%

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

14. Final simplification 18.9%

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

Alternative 4: 18.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 18.8

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in hi around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. *-inverses N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-in N/A

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

   8. mul-1-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

8. Applied rewrites 18.8%

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

10. Applied rewrites 18.8%

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

Alternative 5: 18.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 18.8

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in hi around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. *-inverses N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-in N/A

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

   8. mul-1-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

8. Applied rewrites 18.8%

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

10. Applied rewrites 18.8%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 18.8%

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

Alternative 6: 18.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 18.8

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

5. Applied rewrites 18.8%

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

Alternative 7: 18.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in hi around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 18.8

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

5. Applied rewrites 18.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 18.8%

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

Alternative 8: 18.7% accurate, 18.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lo_, hi_, x_] := 1.0

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in lo around inf

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

5. Applied rewrites 18.7%

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

Reproduce

herbie shell --seed 2024232 
(FPCore (lo hi x)
  :name "xlohi (overflows)"
  :precision binary64
  :pre (and (< lo -1e+308) (> hi 1e+308))
  (/ (- x lo) (- hi lo)))