Kahan p13 Example 1

Percentage Accurate: 99.9% → 99.9%

Time: 7.2s

Alternatives: 10

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t)
use fmin_fmax_functions
    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: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.04938271604938271}{t} + \mathsf{fma}\left(-0.2222222222222222, t, 0.037037037037037035\right)}{t \cdot t} + 0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (fma (fma -2.0 t 1.0) (* t t) 0.5)
     (+
      (/
       (+
        (/ 0.04938271604938271 t)
        (fma -0.2222222222222222 t 0.037037037037037035))
       (* t t))
      0.8333333333333334))))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = fma(fma(-2.0, t, 1.0), (t * t), 0.5);
	} else {
		tmp = (((0.04938271604938271 / t) + fma(-0.2222222222222222, t, 0.037037037037037035)) / (t * t)) + 0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(fma(-2.0, t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(0.04938271604938271 / t) + fma(-0.2222222222222222, t, 0.037037037037037035)) / Float64(t * t)) + 0.8333333333333334);
	end
	return tmp
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]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(N[(-2.0 * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(N[(0.04938271604938271 / t), $MachinePrecision] + N[(-0.2222222222222222 * t + 0.037037037037037035), $MachinePrecision]), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision] + 0.8333333333333334), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))))) < 0.599999999999999978

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. +-commutative N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, 0.5\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)} \]

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right) \]

   if 0.599999999999999978 < (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))))

   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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.2%

      \[\leadsto \color{blue}{0.8333333333333334} \]

   8. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

   10. Applied rewrites 99.3%

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

Alternative 3: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (fma (fma -2.0 t 1.0) (* t t) 0.5)
     (-
      0.8333333333333334
      (/ (- 0.2222222222222222 (/ 0.037037037037037035 t)) t)))))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = fma(fma(-2.0, t, 1.0), (t * t), 0.5);
	} else {
		tmp = 0.8333333333333334 - ((0.2222222222222222 - (0.037037037037037035 / t)) / t);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(fma(-2.0, t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(Float64(0.2222222222222222 - Float64(0.037037037037037035 / t)) / t));
	end
	return tmp
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]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(N[(-2.0 * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(N[(0.2222222222222222 - N[(0.037037037037037035 / t), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))))) < 0.599999999999999978

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. +-commutative N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, 0.5\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)} \]

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right) \]

   if 0.599999999999999978 < (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))))

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\frac{5}{6} + \frac{\frac{1}{27}}{{t}^{2}}\right) - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. associate--r- N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

         \[\leadsto \frac{5}{6} - \left(\frac{\frac{2}{9}}{t} - \color{blue}{\frac{\frac{\frac{1}{27}}{t}}{t}}\right) \]

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

          \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}} \]

      12. *-lft-identity N/A

          \[\leadsto \frac{5}{6} - \color{blue}{1 \cdot \frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{5}{6} - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t} \]

      14. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\frac{5}{6} + -1 \cdot \frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}} \]

      15. *-commutative N/A

          \[\leadsto \frac{5}{6} + \color{blue}{\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t} \cdot -1} \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \color{blue}{\frac{5}{6} - \left(\mathsf{neg}\left(\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}\right)\right) \cdot -1} \]

      17. distribute-lft-neg-out N/A

          \[\leadsto \frac{5}{6} - \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t} \cdot -1\right)\right)} \]

      18. distribute-rgt-neg-in N/A

          \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t} \cdot \left(\mathsf{neg}\left(-1\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \frac{5}{6} - \frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t} \cdot \color{blue}{1} \]

      20. *-rgt-identity N/A

          \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} - \frac{1}{27} \cdot \frac{1}{t}}{t}} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222 - \frac{0.037037037037037035}{t}}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (fma (fma -2.0 t 1.0) (* t t) 0.5)
     (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = fma(fma(-2.0, t, 1.0), (t * t), 0.5);
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(fma(-2.0, t, 1.0), Float64(t * t), 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
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]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(N[(-2.0 * t + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))))) < 0.599999999999999978

   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. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. +-commutative N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), \color{blue}{t \cdot t}, 0.5\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t - 2, t, 1\right), t \cdot t, 0.5\right)} \]

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, t, 1\right), t \cdot t, 0.5\right) \]

   if 0.599999999999999978 < (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))))

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9}}}{t} \]

      4. lower-/.f64 98.5

         \[\leadsto 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222}{t}} \]

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6)
     (fma t t 0.5)
     (- 0.8333333333333334 (/ 0.2222222222222222 t)))))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = 0.8333333333333334 - (0.2222222222222222 / t);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(t, t, 0.5);
	else
		tmp = Float64(0.8333333333333334 - Float64(0.2222222222222222 / t));
	end
	return tmp
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]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(t * t + 0.5), $MachinePrecision], N[(0.8333333333333334 - N[(0.2222222222222222 / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))))) < 0.599999999999999978

   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. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{t}^{2} + \frac{1}{2}} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{t \cdot t} + \frac{1}{2} \]

      3. lower-fma.f64 99.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   if 0.599999999999999978 < (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))))

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{5}{6} - \frac{2}{9} \cdot \frac{1}{t}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{5}{6} - \color{blue}{\frac{\frac{2}{9} \cdot 1}{t}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{5}{6} - \frac{\color{blue}{\frac{2}{9}}}{t} \]

      4. lower-/.f64 98.5

         \[\leadsto 0.8333333333333334 - \color{blue}{\frac{0.2222222222222222}{t}} \]

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{0.8333333333333334 - \frac{0.2222222222222222}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(t, t, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.6) (fma t t 0.5) 0.8333333333333334)))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.6) {
		tmp = fma(t, t, 0.5);
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.6)
		tmp = fma(t, t, 0.5);
	else
		tmp = 0.8333333333333334;
	end
	return tmp
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]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.6], N[(t * t + 0.5), $MachinePrecision], 0.8333333333333334]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))))) < 0.599999999999999978

   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. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {t}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{t}^{2} + \frac{1}{2}} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{t \cdot t} + \frac{1}{2} \]

      3. lower-fma.f64 99.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, t, 0.5\right)} \]

   if 0.599999999999999978 < (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))))

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.2%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ t_2 := t\_1 \cdot t\_1\\ \mathbf{if}\;\frac{1 + t\_2}{2 + t\_2} \leq 0.66:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))) (t_2 (* t_1 t_1)))
   (if (<= (/ (+ 1.0 t_2) (+ 2.0 t_2)) 0.66) 0.5 0.8333333333333334)))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	double t_2 = t_1 * t_1;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (2.0d0 * t) / (1.0d0 + t)
    t_2 = t_1 * t_1
    if (((1.0d0 + t_2) / (2.0d0 + t_2)) <= 0.66d0) then
        tmp = 0.5d0
    else
        tmp = 0.8333333333333334d0
    end if
    code = tmp
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;
	double tmp;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.66) {
		tmp = 0.5;
	} else {
		tmp = 0.8333333333333334;
	}
	return tmp;
}

Python

def code(t):
	t_1 = (2.0 * t) / (1.0 + t)
	t_2 = t_1 * t_1
	tmp = 0
	if ((1.0 + t_2) / (2.0 + t_2)) <= 0.66:
		tmp = 0.5
	else:
		tmp = 0.8333333333333334
	return tmp

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	t_2 = Float64(t_1 * t_1)
	tmp = 0.0
	if (Float64(Float64(1.0 + t_2) / Float64(2.0 + t_2)) <= 0.66)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	return tmp
end

MATLAB

function tmp_2 = code(t)
	t_1 = (2.0 * t) / (1.0 + t);
	t_2 = t_1 * t_1;
	tmp = 0.0;
	if (((1.0 + t_2) / (2.0 + t_2)) <= 0.66)
		tmp = 0.5;
	else
		tmp = 0.8333333333333334;
	end
	tmp_2 = tmp;
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]}, If[LessEqual[N[(N[(1.0 + t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], 0.66], 0.5, 0.8333333333333334]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t))))) < 0.660000000000000031

   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. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{0.5} \]

   if 0.660000000000000031 < (/.f64 (+.f64 #s(literal 1 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))) (+.f64 #s(literal 2 binary64) (*.f64 (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)) (/.f64 (*.f64 #s(literal 2 binary64) t) (+.f64 #s(literal 1 binary64) t)))))

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{5}{6}} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.2%

      \[\leadsto \color{blue}{0.8333333333333334} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\frac{\frac{-4 \cdot t}{t - -1}}{t - -1}\right|\\ t_2 := \mathsf{fma}\left(\frac{t}{-1 - t}, -4 \cdot \frac{t}{t + 1}, 1\right)\\ \mathbf{if}\;t \leq -2 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(-t, t\_1, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(t, t\_1, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (fabs (/ (/ (* -4.0 t) (- t -1.0)) (- t -1.0))))
        (t_2 (fma (/ t (- -1.0 t)) (* -4.0 (/ t (+ t 1.0))) 1.0)))
   (if (<= t -2e-8) (/ t_2 (fma (- t) t_1 2.0)) (/ t_2 (fma t t_1 2.0)))))

C

double code(double t) {
	double t_1 = fabs((((-4.0 * t) / (t - -1.0)) / (t - -1.0)));
	double t_2 = fma((t / (-1.0 - t)), (-4.0 * (t / (t + 1.0))), 1.0);
	double tmp;
	if (t <= -2e-8) {
		tmp = t_2 / fma(-t, t_1, 2.0);
	} else {
		tmp = t_2 / fma(t, t_1, 2.0);
	}
	return tmp;
}

Julia

function code(t)
	t_1 = abs(Float64(Float64(Float64(-4.0 * t) / Float64(t - -1.0)) / Float64(t - -1.0)))
	t_2 = fma(Float64(t / Float64(-1.0 - t)), Float64(-4.0 * Float64(t / Float64(t + 1.0))), 1.0)
	tmp = 0.0
	if (t <= -2e-8)
		tmp = Float64(t_2 / fma(Float64(-t), t_1, 2.0));
	else
		tmp = Float64(t_2 / fma(t, t_1, 2.0));
	end
	return tmp
end

Wolfram

code[t_] := Block[{t$95$1 = N[Abs[N[(N[(N[(-4.0 * t), $MachinePrecision] / N[(t - -1.0), $MachinePrecision]), $MachinePrecision] / N[(t - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * N[(-4.0 * N[(t / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t, -2e-8], N[(t$95$2 / N[((-t) * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(t * t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < -2e-8

   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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 5.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lift--.f64 5.7

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

   8. Applied rewrites 5.7%

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

      1. unpow1 N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unpow-prod-down N/A

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

      8. metadata-eval N/A

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

      9. sqrt-pow1 N/A

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

      10. pow2 N/A

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

      11. sqr-neg-rev N/A

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

      12. pow2 N/A

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

      13. sqrt-pow1 N/A

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

      14. metadata-eval N/A

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

      15. unpow1 N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 99.9

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

   10. Applied rewrites 99.9%

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

   if -2e-8 < t

   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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lift--.f64 100.0

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

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{t}{-1 - t}}, -4 \cdot \frac{t}{t + 1}, 1\right)}{\mathsf{fma}\left(t, \left|\frac{\frac{-4 \cdot t}{-1 - t}}{-1 - t}\right|, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t}{-1 - t}, -4 \cdot \frac{t}{t + 1}, 1\right)}{\mathsf{fma}\left(-t, \left|\frac{\frac{-4 \cdot t}{t - -1}}{t - -1}\right|, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t}{-1 - t}, -4 \cdot \frac{t}{t + 1}, 1\right)}{\mathsf{fma}\left(t, \left|\frac{\frac{-4 \cdot t}{t - -1}}{t - -1}\right|, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 \cdot t}{1 + t}\\ \frac{\mathsf{fma}\left(\frac{t}{\mathsf{fma}\left(-1, t, -1\right)}, -4 \cdot \frac{t}{t + 1}, 1\right)}{2 + t\_1 \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (/ (* 2.0 t) (+ 1.0 t))))
   (/
    (fma (/ t (fma -1.0 t -1.0)) (* -4.0 (/ t (+ t 1.0))) 1.0)
    (+ 2.0 (* t_1 t_1)))))

C

double code(double t) {
	double t_1 = (2.0 * t) / (1.0 + t);
	return fma((t / fma(-1.0, t, -1.0)), (-4.0 * (t / (t + 1.0))), 1.0) / (2.0 + (t_1 * t_1));
}

Julia

function code(t)
	t_1 = Float64(Float64(2.0 * t) / Float64(1.0 + t))
	return Float64(fma(Float64(t / fma(-1.0, t, -1.0)), Float64(-4.0 * Float64(t / Float64(t + 1.0))), 1.0) / Float64(2.0 + Float64(t_1 * t_1)))
end

Wolfram

code[t_] := Block[{t$95$1 = N[(N[(2.0 * t), $MachinePrecision] / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t / N[(-1.0 * t + -1.0), $MachinePrecision]), $MachinePrecision] * N[(-4.0 * N[(t / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(2.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $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

4. Applied rewrites 100.0%

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

Alternative 10: 58.7% accurate, 104.0× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t)
use fmin_fmax_functions
    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

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. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 60.3%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024351 
(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))))))