Kahan p13 Example 2

Percentage Accurate: 99.9% → 100.0%

Time: 17.2s

Alternatives: 7

Speedup: 1.9×

Specification

Math

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 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 - ((2.0 / t) / (1.0 + (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 - ((2.0d0 / t) / (1.0d0 + (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 - ((2.0 / t) / (1.0 + (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 - ((2.0 / t) / (1.0 + (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(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + 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 - ((2.0 / t) / (1.0 + (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[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $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

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

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 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 - ((2.0 / t) / (1.0 + (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 - ((2.0d0 / t) / (1.0d0 + (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 - ((2.0 / t) / (1.0 + (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 - ((2.0 / t) / (1.0 + (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(2.0 - Float64(Float64(2.0 / t) / Float64(1.0 + 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 - ((2.0 / t) / (1.0 + (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[(2.0 - N[(N[(2.0 / t), $MachinePrecision] / N[(1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $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: 100.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{5 + \frac{8 + \frac{-4}{t + 1}}{-1 - t}}{6 + \frac{-2}{\frac{t + 1}{\frac{-2}{t + 1} + 4}}} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (/
  (+ 5.0 (/ (+ 8.0 (/ -4.0 (+ t 1.0))) (- -1.0 t)))
  (+ 6.0 (/ -2.0 (/ (+ t 1.0) (+ (/ -2.0 (+ t 1.0)) 4.0))))))

C

double code(double t) {
	return (5.0 + ((8.0 + (-4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + (-2.0 / ((t + 1.0) / ((-2.0 / (t + 1.0)) + 4.0))));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = (5.0d0 + ((8.0d0 + ((-4.0d0) / (t + 1.0d0))) / ((-1.0d0) - t))) / (6.0d0 + ((-2.0d0) / ((t + 1.0d0) / (((-2.0d0) / (t + 1.0d0)) + 4.0d0))))
end function

Java

public static double code(double t) {
	return (5.0 + ((8.0 + (-4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + (-2.0 / ((t + 1.0) / ((-2.0 / (t + 1.0)) + 4.0))));
}

Python

def code(t):
	return (5.0 + ((8.0 + (-4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + (-2.0 / ((t + 1.0) / ((-2.0 / (t + 1.0)) + 4.0))))

Julia

function code(t)
	return Float64(Float64(5.0 + Float64(Float64(8.0 + Float64(-4.0 / Float64(t + 1.0))) / Float64(-1.0 - t))) / Float64(6.0 + Float64(-2.0 / Float64(Float64(t + 1.0) / Float64(Float64(-2.0 / Float64(t + 1.0)) + 4.0)))))
end

MATLAB

function tmp = code(t)
	tmp = (5.0 + ((8.0 + (-4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + (-2.0 / ((t + 1.0) / ((-2.0 / (t + 1.0)) + 4.0))));
end

Wolfram

code[t_] := N[(N[(5.0 + N[(N[(8.0 + N[(-4.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(-2.0 / N[(N[(t + 1.0), $MachinePrecision] / N[(N[(-2.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 2: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{5 + \frac{8 + \frac{-4}{t + 1}}{-1 - t}}{6 + \frac{-8 + \frac{4}{t + 1}}{t + 1}} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (/
  (+ 5.0 (/ (+ 8.0 (/ -4.0 (+ t 1.0))) (- -1.0 t)))
  (+ 6.0 (/ (+ -8.0 (/ 4.0 (+ t 1.0))) (+ t 1.0)))))

C

double code(double t) {
	return (5.0 + ((8.0 + (-4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + ((-8.0 + (4.0 / (t + 1.0))) / (t + 1.0)));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = (5.0d0 + ((8.0d0 + ((-4.0d0) / (t + 1.0d0))) / ((-1.0d0) - t))) / (6.0d0 + (((-8.0d0) + (4.0d0 / (t + 1.0d0))) / (t + 1.0d0)))
end function

Java

public static double code(double t) {
	return (5.0 + ((8.0 + (-4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + ((-8.0 + (4.0 / (t + 1.0))) / (t + 1.0)));
}

Python

def code(t):
	return (5.0 + ((8.0 + (-4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + ((-8.0 + (4.0 / (t + 1.0))) / (t + 1.0)))

Julia

function code(t)
	return Float64(Float64(5.0 + Float64(Float64(8.0 + Float64(-4.0 / Float64(t + 1.0))) / Float64(-1.0 - t))) / Float64(6.0 + Float64(Float64(-8.0 + Float64(4.0 / Float64(t + 1.0))) / Float64(t + 1.0))))
end

MATLAB

function tmp = code(t)
	tmp = (5.0 + ((8.0 + (-4.0 / (t + 1.0))) / (-1.0 - t))) / (6.0 + ((-8.0 + (4.0 / (t + 1.0))) / (t + 1.0)));
end

Wolfram

code[t_] := N[(N[(5.0 + N[(N[(8.0 + N[(-4.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(6.0 + N[(N[(-8.0 + N[(4.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 3: 99.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot 2 - 2\\ \mathbf{if}\;t \leq -0.6 \lor \neg \left(t \leq 0.48\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{5 + t_1 \cdot \left(t_1 - -4\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (- (* t 2.0) 2.0)))
   (if (or (<= t -0.6) (not (<= t 0.48)))
     (- 0.8333333333333334 (/ 0.2222222222222222 t))
     (/ (+ 5.0 (* t_1 (- t_1 -4.0))) 2.0))))

C

double code(double t) {
	double t_1 = (t * 2.0) - 2.0;
	double tmp;
	if ((t <= -0.6) || !(t <= 0.48)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (5.0 + (t_1 * (t_1 - -4.0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t * 2.0d0) - 2.0d0
    if ((t <= (-0.6d0)) .or. (.not. (t <= 0.48d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = (5.0d0 + (t_1 * (t_1 - (-4.0d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double t_1 = (t * 2.0) - 2.0;
	double tmp;
	if ((t <= -0.6) || !(t <= 0.48)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = (5.0 + (t_1 * (t_1 - -4.0))) / 2.0;
	}
	return tmp;
}

Python

def code(t):
	t_1 = (t * 2.0) - 2.0
	tmp = 0
	if (t <= -0.6) or not (t <= 0.48):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = (5.0 + (t_1 * (t_1 - -4.0))) / 2.0
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(t * 2.0) - 2.0)
	tmp = 0.0
	if ((t <= -0.6) || !(t <= 0.48))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = Float64(Float64(5.0 + Float64(t_1 * Float64(t_1 - -4.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (t * 2.0) - 2.0;
	tmp = 0.0;
	if ((t <= -0.6) || ~((t <= 0.48)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = (5.0 + (t_1 * (t_1 - -4.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(t * 2.0), $MachinePrecision] - 2.0), $MachinePrecision]}, If[Or[LessEqual[t, -0.6], N[Not[LessEqual[t, 0.48]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], N[(N[(5.0 + N[(t$95$1 * N[(t$95$1 - -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 4: 97.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-2}{t + 1} \cdot 2\\ \frac{5 + t_1}{6 + t_1} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (* (/ -2.0 (+ t 1.0)) 2.0))) (/ (+ 5.0 t_1) (+ 6.0 t_1))))

C

double code(double t) {
	double t_1 = (-2.0 / (t + 1.0)) * 2.0;
	return (5.0 + t_1) / (6.0 + t_1);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = ((-2.0d0) / (t + 1.0d0)) * 2.0d0
    code = (5.0d0 + t_1) / (6.0d0 + t_1)
end function

Java

public static double code(double t) {
	double t_1 = (-2.0 / (t + 1.0)) * 2.0;
	return (5.0 + t_1) / (6.0 + t_1);
}

Python

def code(t):
	t_1 = (-2.0 / (t + 1.0)) * 2.0
	return (5.0 + t_1) / (6.0 + t_1)

Julia

function code(t)
	t_1 = Float64(Float64(-2.0 / Float64(t + 1.0)) * 2.0)
	return Float64(Float64(5.0 + t_1) / Float64(6.0 + t_1))
end

MATLAB

function tmp = code(t)
	t_1 = (-2.0 / (t + 1.0)) * 2.0;
	tmp = (5.0 + t_1) / (6.0 + t_1);
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(-2.0 / N[(t + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, N[(N[(5.0 + t$95$1), $MachinePrecision] / N[(6.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 5: 99.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.49 \lor \neg \left(t \leq 0.66\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (or (<= t -0.49) (not (<= t 0.66)))
   (- 0.8333333333333334 (/ 0.2222222222222222 t))
   0.5))

C

double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.66)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.49d0)) .or. (.not. (t <= 0.66d0))) then
        tmp = 0.8333333333333334d0 - (0.2222222222222222d0 / t)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if ((t <= -0.49) || !(t <= 0.66)) {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if (t <= -0.49) or not (t <= 0.66):
		tmp = 0.8333333333333334 - (0.2222222222222222 / t)
	else:
		tmp = 0.5
	return tmp

Julia

function code(t)
	tmp = 0.0
	if ((t <= -0.49) || !(t <= 0.66))
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if ((t <= -0.49) || ~((t <= 0.66)))
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[Or[LessEqual[t, -0.49], N[Not[LessEqual[t, 0.66]], $MachinePrecision]], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

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1. Add Preprocessing

Alternative 6: 98.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.34:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (if (<= t -0.34) 0.8333333333333334 (if (<= t 1.0) 0.5 0.8333333333333334)))

C

double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-0.34d0)) then
        tmp = 0.8333333333333334d0
    else if (t <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
end function

Java

public static double code(double t) {
	double tmp;
	if (t <= -0.34) {
		tmp = 0.8333333333333334;
	} else if (t <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	tmp = 0
	if t <= -0.34:
		tmp = 0.8333333333333334
	elif t <= 1.0:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	tmp = 0.0
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	tmp = 0.0;
	if (t <= -0.34)
		tmp = 0.8333333333333334;
	elseif (t <= 1.0)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
end

Wolfram

code[t_] := If[LessEqual[t, -0.34], 0.8333333333333334, If[LessEqual[t, 1.0], 0.5, 0.8333333333333334]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 59.2% accurate, 51.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (t) :precision binary64 0.5)

C

double code(double t) {
	return 0.5;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 0.5d0
end function

Java

public static double code(double t) {
	return 0.5;
}

Python

def code(t):
	return 0.5

Julia

function code(t)
	return 0.5
end

MATLAB

function tmp = code(t)
	tmp = 0.5;
end

Wolfram

code[t_] := 0.5

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Reproduce

herbie shell --seed 2023350 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))