Kahan p13 Example 1

Percentage Accurate: 99.9% → 99.9%

Time: 10.5s

Alternatives: 13

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = (2.0d0 * t) / (1.0d0 + t)
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = (2.0 * t) / (1.0 + t)
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = (2.0 * t) / (1.0 + t);
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = (2.0d0 * t) / (1.0d0 + t)
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = (2.0 * t) / (1.0 + t)
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = (2.0 * t) / (1.0 + t);
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \frac{1 + t\_2}{2 + t\_2} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (/ (+ 1.0 t_2) (+ 2.0 t_2))))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = (2.0d0 * t) / (1.0d0 + t)
    t_2 = t_1 * t_1
    code = (1.0d0 + t_2) / (2.0d0 + t_2)
end function

Java

public static double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	return (1.0 + t_2) / (2.0 + t_2);
}

Python

def code(t):
	t_1 = (2.0 * t) / (1.0 + t)
	t_2 = t_1 * t_1
	return (1.0 + t_2) / (2.0 + t_2)

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2))
end

MATLAB

function tmp = code(t)
	t_1 = (2.0 * t) / (1.0 + t);
	t_2 = t_1 * t_1;
	tmp = (1.0 + t_2) / (2.0 + t_2);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 100.0%

   \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}\\ \mathbf{if}\;\frac{2 \cdot t}{1 + t} \leq 1.99999:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, t\_1, 1\right)}{\mathsf{fma}\left(t, t\_1, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right) - \frac{-0.037037037037037035 + \frac{-0.04938271604938271}{t}}{t \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* t 4.0) (fma t (+ 2.0 t) 1.0))))
   (if (<= (/ (* 2.0 t) (+ 1.0 t)) 1.99999)
     (/ (fma t t_1 1.0) (fma t t_1 2.0))
     (-
      (- 0.8333333333333334 (/ 0.2222222222222222 t))
      (/ (+ -0.037037037037037035 (/ -0.04938271604938271 t)) (* t t))))))

C

double code(double t) {
	double t_1 = (t * 4.0) / fma(t, (2.0 + t), 1.0);
	double tmp;
	if (((2.0 * t) / (1.0 + t)) <= 1.99999) {
		tmp = fma(t, t_1, 1.0) / fma(t, t_1, 2.0);
	} else {
		tmp = (0.8333333333333334 - (0.2222222222222222 / t)) - ((-0.037037037037037035 + (-0.04938271604938271 / t)) / (t * t));
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(Float64(t * 4.0) / fma(t, Float64(2.0 + t), 1.0))
	tmp = 0.0
	if (Float64(Float64(2.0 * t) / Float64(1.0 + t)) <= 1.99999)
		tmp = Float64(fma(t, t_1, 1.0) / fma(t, t_1, 2.0));
	else
		tmp = Float64(Float64(0.8333333333333334 - Float64(0.2222222222222222 / t)) - Float64(Float64(-0.037037037037037035 + Float64(-0.04938271604938271 / t)) / Float64(t * t)));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(t * 4.0), $MachinePrecision] / N[(t * N[(2.0 + t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision], 1.99999], N[(N[(t * t$95$1 + 1.0), $MachinePrecision] / N[(t * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.037037037037037035 + N[(-0.04938271604938271 / t), $MachinePrecision]), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) < 1.9999899999999999

   1. Initial program 100.0%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}}}{2 + \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)} + 1}}{2 + \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}} + 1}{2 + \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{t \cdot \left(t \cdot 4\right)}}{\left(1 + t\right) \cdot \left(1 + t\right)} + 1}{2 + \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{t \cdot \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}} + 1}{2 + \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}}{2 + \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}} \]

      7. lower-/.f64 100.0

         \[\leadsto \frac{\mathsf{fma}\left(t, \color{blue}{\frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}}, 1\right)}{2 + \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\color{blue}{2 + \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\color{blue}{\frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)} + 2}} \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\color{blue}{\frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}} + 2} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\frac{\color{blue}{t \cdot \left(t \cdot 4\right)}}{\left(1 + t\right) \cdot \left(1 + t\right)} + 2} \]

      12. associate-/l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\color{blue}{t \cdot \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}} + 2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\color{blue}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)}} \]

      14. lower-/.f64 100.0

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \color{blue}{\frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}}, 2\right)} \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 2\right)}} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\color{blue}{1 + t \cdot \left(2 + t\right)}}, 2\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\color{blue}{t \cdot \left(2 + t\right) + 1}}, 2\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\color{blue}{\left(2 \cdot t + t \cdot t\right)} + 1}, 2\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(2 \cdot t + \color{blue}{{t}^{2}}\right) + 1}, 2\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(\color{blue}{t \cdot 2} + {t}^{2}\right) + 1}, 2\right)} \]

      5. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(\color{blue}{\left(1 \cdot t\right)} \cdot 2 + {t}^{2}\right) + 1}, 2\right)} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(\left(\color{blue}{\left(\frac{1}{t} \cdot t\right)} \cdot t\right) \cdot 2 + {t}^{2}\right) + 1}, 2\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(\color{blue}{\left(\frac{1}{t} \cdot \left(t \cdot t\right)\right)} \cdot 2 + {t}^{2}\right) + 1}, 2\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(\left(\frac{1}{t} \cdot \color{blue}{{t}^{2}}\right) \cdot 2 + {t}^{2}\right) + 1}, 2\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(\color{blue}{\left({t}^{2} \cdot \frac{1}{t}\right)} \cdot 2 + {t}^{2}\right) + 1}, 2\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(\color{blue}{{t}^{2} \cdot \left(\frac{1}{t} \cdot 2\right)} + {t}^{2}\right) + 1}, 2\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left({t}^{2} \cdot \color{blue}{\left(2 \cdot \frac{1}{t}\right)} + {t}^{2}\right) + 1}, 2\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(\color{blue}{\left(2 \cdot \frac{1}{t}\right) \cdot {t}^{2}} + {t}^{2}\right) + 1}, 2\right)} \]

      13. *-lft-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(\left(2 \cdot \frac{1}{t}\right) \cdot {t}^{2} + \color{blue}{1 \cdot {t}^{2}}\right) + 1}, 2\right)} \]

      14. distribute-rgt-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\color{blue}{{t}^{2} \cdot \left(2 \cdot \frac{1}{t} + 1\right)} + 1}, 2\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{{t}^{2} \cdot \color{blue}{\left(1 + 2 \cdot \frac{1}{t}\right)} + 1}, 2\right)} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\color{blue}{\left(t \cdot t\right)} \cdot \left(1 + 2 \cdot \frac{1}{t}\right) + 1}, 2\right)} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\color{blue}{t \cdot \left(t \cdot \left(1 + 2 \cdot \frac{1}{t}\right)\right)} + 1}, 2\right)} \]

      18. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\color{blue}{\mathsf{fma}\left(t, t \cdot \left(1 + 2 \cdot \frac{1}{t}\right), 1\right)}}, 2\right)} \]

   9. Applied rewrites 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\left(1 + t\right) \cdot \left(1 + t\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\color{blue}{\mathsf{fma}\left(t, 2 + t, 1\right)}}, 2\right)} \]

   10. Taylor expanded in t around 0

       \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\color{blue}{1 + t \cdot \left(2 + t\right)}}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\color{blue}{t \cdot \left(2 + t\right) + 1}}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

       2. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{t \cdot \left(\color{blue}{2 \cdot 1} + t\right) + 1}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

       3. lft-mult-inverse N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{t \cdot \left(2 \cdot \color{blue}{\left(\frac{1}{t} \cdot t\right)} + t\right) + 1}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

       4. associate-*l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{t \cdot \left(\color{blue}{\left(2 \cdot \frac{1}{t}\right) \cdot t} + t\right) + 1}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

       5. distribute-lft1-in N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{t \cdot \color{blue}{\left(\left(2 \cdot \frac{1}{t} + 1\right) \cdot t\right)} + 1}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

       6. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{t \cdot \left(\color{blue}{\left(1 + 2 \cdot \frac{1}{t}\right)} \cdot t\right) + 1}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{t \cdot \color{blue}{\left(t \cdot \left(1 + 2 \cdot \frac{1}{t}\right)\right)} + 1}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{t \cdot \color{blue}{\left(\left(1 + 2 \cdot \frac{1}{t}\right) \cdot t\right)} + 1}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\color{blue}{\mathsf{fma}\left(t, \left(1 + 2 \cdot \frac{1}{t}\right) \cdot t, 1\right)}}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

       10. +-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, \color{blue}{\left(2 \cdot \frac{1}{t} + 1\right)} \cdot t, 1\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

       11. distribute-lft1-in N/A

           \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, \color{blue}{\left(2 \cdot \frac{1}{t}\right) \cdot t + t}, 1\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

       12. associate-*l* N/A

           \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, \color{blue}{2 \cdot \left(\frac{1}{t} \cdot t\right)} + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

       13. lft-mult-inverse N/A

           \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 \cdot \color{blue}{1} + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

       14. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, \color{blue}{2} + t, 1\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

       15. lower-+.f64 100.0

           \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, \color{blue}{2 + t}, 1\right)}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

   12. Applied rewrites 100.0%

       \[\leadsto \frac{\mathsf{fma}\left(t, \frac{t \cdot 4}{\color{blue}{\mathsf{fma}\left(t, 2 + t, 1\right)}}, 1\right)}{\mathsf{fma}\left(t, \frac{t \cdot 4}{\mathsf{fma}\left(t, 2 + t, 1\right)}, 2\right)} \]

   if 1.9999899999999999 < (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))

   1. Initial program 99.9%

      \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\frac{1 + \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}}{2 + \frac{t \cdot \left(t \cdot 4\right)}{\left(1 + t\right) \cdot \left(1 + t\right)}}} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \left(\frac{\frac{1}{27}}{{t}^{2}} + \frac{4}{81} \cdot \frac{1}{{t}^{3}}\right)\right) - \frac{2}{9} \cdot \frac{1}{t}} \]

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right) - \frac{-0.037037037037037035 + \frac{-0.04938271604938271}{t}}{t \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t)))) (+ 2.0 (* (/ (* 2.0 t) (+ 1.0 t)) (/ (* 2.0 t) (+ 1.0 t))))))