Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1

Percentage Accurate: 27.7% → 30.8%

Time: 6.8s

Alternatives: 5

Speedup: 2.6×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (*
  (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
  (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t) / 16.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

Python

def code(x, y, z, t, a, b):
	return (x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 27.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (*
  (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
  (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t) / 16.0d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

Python

def code(x, y, z, t, a, b):
	return (x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 30.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \sin \left(0.5 \cdot \pi\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* (sin (* 0.5 PI)) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return sin((0.5 * ((double) M_PI))) * x;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return Math.sin((0.5 * Math.PI)) * x;
}

Python

def code(x, y, z, t, a, b):
	return math.sin((0.5 * math.pi)) * x

Julia

function code(x, y, z, t, a, b)
	return Float64(sin(Float64(0.5 * pi)) * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = sin((0.5 * pi)) * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 27.7%

   \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \color{blue}{x} \]

   2. lower-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \color{blue}{x} \]

   3. lower-cos.f64 N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x \]

   4. *-commutative N/A

      \[\leadsto \cos \left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right) \cdot x \]

   5. lower-*.f64 N/A

      \[\leadsto \cos \left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right) \cdot x \]

   6. *-commutative N/A

      \[\leadsto \cos \left(\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   7. lower-*.f64 N/A

      \[\leadsto \cos \left(\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   8. *-commutative N/A

      \[\leadsto \cos \left(\left(\left(\left(1 + 2 \cdot a\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   9. lower-*.f64 N/A

      \[\leadsto \cos \left(\left(\left(\left(1 + 2 \cdot a\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   10. +-commutative N/A

       \[\leadsto \cos \left(\left(\left(\left(2 \cdot a + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   11. *-commutative N/A

       \[\leadsto \cos \left(\left(\left(\left(a \cdot 2 + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   12. lower-fma.f64 29.0

       \[\leadsto \cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x \]

4. Applied rewrites 29.0%

   \[\leadsto \color{blue}{\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x} \]

5. Taylor expanded in a around 0

   \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]
6. Step-by-step derivation

   1. lower-cos.f64 N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]

   2. *-commutative N/A

      \[\leadsto \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \cdot x \]

   3. lift-*.f64 N/A

      \[\leadsto \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \cdot x \]

   4. lift-*.f64 29.9

      \[\leadsto \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot x \]

7. Applied rewrites 29.9%

   \[\leadsto \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot x \]
8. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \cdot x \]

   2. sin-+PI/2-rev N/A

      \[\leadsto \sin \left(\left(b \cdot t\right) \cdot \frac{1}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot x \]

   3. lift-*.f64 N/A

      \[\leadsto \sin \left(\left(b \cdot t\right) \cdot \frac{1}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot x \]

   4. lift-*.f64 N/A

      \[\leadsto \sin \left(\left(b \cdot t\right) \cdot \frac{1}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot x \]

   5. *-commutative N/A

      \[\leadsto \sin \left(\frac{1}{16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot x \]

   6. lower-sin.f64 N/A

      \[\leadsto \sin \left(\frac{1}{16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot x \]

   7. *-commutative N/A

      \[\leadsto \sin \left(\left(b \cdot t\right) \cdot \frac{1}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot x \]

   8. lower-fma.f64 N/A

      \[\leadsto \sin \left(\mathsf{fma}\left(b \cdot t, \frac{1}{16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot x \]

   9. lift-*.f64 N/A

      \[\leadsto \sin \left(\mathsf{fma}\left(b \cdot t, \frac{1}{16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot x \]

   10. lower-/.f64 N/A

       \[\leadsto \sin \left(\mathsf{fma}\left(b \cdot t, \frac{1}{16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot x \]

   11. lower-PI.f64 29.9

       \[\leadsto \sin \left(\mathsf{fma}\left(b \cdot t, 0.0625, \frac{\pi}{2}\right)\right) \cdot x \]

9. Applied rewrites 29.9%

   \[\leadsto \sin \left(\mathsf{fma}\left(b \cdot t, 0.0625, \frac{\pi}{2}\right)\right) \cdot x \]

10. Taylor expanded in t around 0

    \[\leadsto \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot x \]

    2. lift-PI.f64 30.8

       \[\leadsto \sin \left(0.5 \cdot \pi\right) \cdot x \]

12. Applied rewrites 30.8%

    \[\leadsto \sin \left(0.5 \cdot \pi\right) \cdot x \]
13. Add Preprocessing

Alternative 2: 26.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot -0.001953125\right) \cdot t, t, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot \left(t \cdot t\right)\right) \cdot b\right) \cdot b, -0.001953125, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 1.5e+150)
   (* (fma (* (* (* b b) -0.001953125) t) t 1.0) x)
   (*
    (fma
     (* (* (* (* (fma a 2.0 1.0) (fma a 2.0 1.0)) (* t t)) b) b)
     -0.001953125
     1.0)
    x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 1.5e+150) {
		tmp = fma((((b * b) * -0.001953125) * t), t, 1.0) * x;
	} else {
		tmp = fma(((((fma(a, 2.0, 1.0) * fma(a, 2.0, 1.0)) * (t * t)) * b) * b), -0.001953125, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 1.5e+150)
		tmp = Float64(fma(Float64(Float64(Float64(b * b) * -0.001953125) * t), t, 1.0) * x);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(fma(a, 2.0, 1.0) * fma(a, 2.0, 1.0)) * Float64(t * t)) * b) * b), -0.001953125, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 1.5e+150], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.001953125), $MachinePrecision] * t), $MachinePrecision] * t + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * -0.001953125 + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.50000000000000006e150

   1. Initial program 27.7%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \color{blue}{x} \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x \]

      4. *-commutative N/A

         \[\leadsto \cos \left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \cos \left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right) \cdot x \]

      6. *-commutative N/A

         \[\leadsto \cos \left(\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

      7. lower-*.f64 N/A

         \[\leadsto \cos \left(\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

      8. *-commutative N/A

         \[\leadsto \cos \left(\left(\left(\left(1 + 2 \cdot a\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

      9. lower-*.f64 N/A

         \[\leadsto \cos \left(\left(\left(\left(1 + 2 \cdot a\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

      10. +-commutative N/A

          \[\leadsto \cos \left(\left(\left(\left(2 \cdot a + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

      11. *-commutative N/A

          \[\leadsto \cos \left(\left(\left(\left(a \cdot 2 + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

      12. lower-fma.f64 29.0

          \[\leadsto \cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x \]

   4. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x} \]

   5. Taylor expanded in a around 0

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. lower-cos.f64 N/A

         \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \cdot x \]

      3. lift-*.f64 N/A

         \[\leadsto \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \cdot x \]

      4. lift-*.f64 29.9

         \[\leadsto \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot x \]

   7. Applied rewrites 29.9%

      \[\leadsto \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot x \]

   8. Taylor expanded in t around 0

      \[\leadsto \left(1 + \frac{-1}{512} \cdot \left({b}^{2} \cdot {t}^{2}\right)\right) \cdot x \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {t}^{2}\right) + 1\right) \cdot x \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{512} \cdot {b}^{2}\right) \cdot {t}^{2} + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{512} \cdot {b}^{2}, {t}^{2}, 1\right) \cdot x \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

      6. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{-1}{512}, t \cdot t, 1\right) \cdot x \]

      9. lift-*.f64 24.7

         \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot -0.001953125, t \cdot t, 1\right) \cdot x \]

   10. Applied rewrites 24.7%

       \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot -0.001953125, t \cdot t, 1\right) \cdot x \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{-1}{512}, t \cdot t, 1\right) \cdot x \]

       2. lift-fma.f64 N/A

          \[\leadsto \left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot \left(t \cdot t\right) + 1\right) \cdot x \]

       3. associate-*r* N/A

          \[\leadsto \left(\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t\right) \cdot t + 1\right) \cdot x \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t, t, 1\right) \cdot x \]

       5. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t, t, 1\right) \cdot x \]

       6. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t, t, 1\right) \cdot x \]

       7. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\left({b}^{2} \cdot \frac{-1}{512}\right) \cdot t, t, 1\right) \cdot x \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{512} \cdot {b}^{2}\right) \cdot t, t, 1\right) \cdot x \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{512} \cdot {b}^{2}\right) \cdot t, t, 1\right) \cdot x \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\left({b}^{2} \cdot \frac{-1}{512}\right) \cdot t, t, 1\right) \cdot x \]

       11. pow2 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t, t, 1\right) \cdot x \]

       12. lift-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t, t, 1\right) \cdot x \]

       13. lift-*.f64 25.6

           \[\leadsto \mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot -0.001953125\right) \cdot t, t, 1\right) \cdot x \]

   12. Applied rewrites 25.6%

       \[\leadsto \mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot -0.001953125\right) \cdot t, t, 1\right) \cdot x \]

   if 1.50000000000000006e150 < b

   1. Initial program 27.7%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \color{blue}{x} \]

      2. lower-*.f64 N/A

         \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \color{blue}{x} \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x \]

      4. *-commutative N/A

         \[\leadsto \cos \left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right) \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \cos \left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right) \cdot x \]

      6. *-commutative N/A

         \[\leadsto \cos \left(\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

      7. lower-*.f64 N/A

         \[\leadsto \cos \left(\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

      8. *-commutative N/A

         \[\leadsto \cos \left(\left(\left(\left(1 + 2 \cdot a\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

      9. lower-*.f64 N/A

         \[\leadsto \cos \left(\left(\left(\left(1 + 2 \cdot a\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

      10. +-commutative N/A

          \[\leadsto \cos \left(\left(\left(\left(2 \cdot a + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

      11. *-commutative N/A

          \[\leadsto \cos \left(\left(\left(\left(a \cdot 2 + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

      12. lower-fma.f64 29.0

          \[\leadsto \cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x \]

   4. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x} \]

   5. Taylor expanded in a around 0

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. lower-cos.f64 N/A

         \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \cdot x \]

      3. lift-*.f64 N/A

         \[\leadsto \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \cdot x \]

      4. lift-*.f64 29.9

         \[\leadsto \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot x \]

   7. Applied rewrites 29.9%

      \[\leadsto \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot x \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \cdot x \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \sin \left(\left(b \cdot t\right) \cdot \frac{1}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot x \]

      3. lift-*.f64 N/A

         \[\leadsto \sin \left(\left(b \cdot t\right) \cdot \frac{1}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot x \]

      4. lift-*.f64 N/A

         \[\leadsto \sin \left(\left(b \cdot t\right) \cdot \frac{1}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \sin \left(\frac{1}{16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot x \]

      6. lower-sin.f64 N/A

         \[\leadsto \sin \left(\frac{1}{16} \cdot \left(b \cdot t\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \sin \left(\left(b \cdot t\right) \cdot \frac{1}{16} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot x \]

      8. lower-fma.f64 N/A

         \[\leadsto \sin \left(\mathsf{fma}\left(b \cdot t, \frac{1}{16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot x \]

      9. lift-*.f64 N/A

         \[\leadsto \sin \left(\mathsf{fma}\left(b \cdot t, \frac{1}{16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot x \]

      10. lower-/.f64 N/A

          \[\leadsto \sin \left(\mathsf{fma}\left(b \cdot t, \frac{1}{16}, \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot x \]

      11. lower-PI.f64 29.9

          \[\leadsto \sin \left(\mathsf{fma}\left(b \cdot t, 0.0625, \frac{\pi}{2}\right)\right) \cdot x \]

   9. Applied rewrites 29.9%

      \[\leadsto \sin \left(\mathsf{fma}\left(b \cdot t, 0.0625, \frac{\pi}{2}\right)\right) \cdot x \]

   10. Taylor expanded in t around 0

       \[\leadsto \left(1 + \frac{-1}{512} \cdot \left({b}^{2} \cdot \left({t}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right)\right)\right) \cdot x \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot \left({t}^{2} \cdot {\left(1 + 2 \cdot a\right)}^{2}\right)\right) + 1\right) \cdot x \]

   12. Applied rewrites 22.4%

       \[\leadsto \mathsf{fma}\left(\left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot \mathsf{fma}\left(a, 2, 1\right)\right) \cdot \left(t \cdot t\right)\right) \cdot b\right) \cdot b, -0.001953125, 1\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 25.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot -0.001953125\right) \cdot t, t, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* (fma (* (* (* b b) -0.001953125) t) t 1.0) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma((((b * b) * -0.001953125) * t), t, 1.0) * x;
}

Julia

function code(x, y, z, t, a, b)
	return Float64(fma(Float64(Float64(Float64(b * b) * -0.001953125) * t), t, 1.0) * x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.001953125), $MachinePrecision] * t), $MachinePrecision] * t + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 27.7%

   \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \color{blue}{x} \]

   2. lower-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \color{blue}{x} \]

   3. lower-cos.f64 N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x \]

   4. *-commutative N/A

      \[\leadsto \cos \left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right) \cdot x \]

   5. lower-*.f64 N/A

      \[\leadsto \cos \left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right) \cdot x \]

   6. *-commutative N/A

      \[\leadsto \cos \left(\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   7. lower-*.f64 N/A

      \[\leadsto \cos \left(\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   8. *-commutative N/A

      \[\leadsto \cos \left(\left(\left(\left(1 + 2 \cdot a\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   9. lower-*.f64 N/A

      \[\leadsto \cos \left(\left(\left(\left(1 + 2 \cdot a\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   10. +-commutative N/A

       \[\leadsto \cos \left(\left(\left(\left(2 \cdot a + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   11. *-commutative N/A

       \[\leadsto \cos \left(\left(\left(\left(a \cdot 2 + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   12. lower-fma.f64 29.0

       \[\leadsto \cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x \]

4. Applied rewrites 29.0%

   \[\leadsto \color{blue}{\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x} \]

5. Taylor expanded in a around 0

   \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]
6. Step-by-step derivation

   1. lower-cos.f64 N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]

   2. *-commutative N/A

      \[\leadsto \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \cdot x \]

   3. lift-*.f64 N/A

      \[\leadsto \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \cdot x \]

   4. lift-*.f64 29.9

      \[\leadsto \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot x \]

7. Applied rewrites 29.9%

   \[\leadsto \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot x \]

8. Taylor expanded in t around 0

   \[\leadsto \left(1 + \frac{-1}{512} \cdot \left({b}^{2} \cdot {t}^{2}\right)\right) \cdot x \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {t}^{2}\right) + 1\right) \cdot x \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-1}{512} \cdot {b}^{2}\right) \cdot {t}^{2} + 1\right) \cdot x \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{512} \cdot {b}^{2}, {t}^{2}, 1\right) \cdot x \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

   6. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

   7. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

   8. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{-1}{512}, t \cdot t, 1\right) \cdot x \]

   9. lift-*.f64 24.7

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot -0.001953125, t \cdot t, 1\right) \cdot x \]

10. Applied rewrites 24.7%

    \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot -0.001953125, t \cdot t, 1\right) \cdot x \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{-1}{512}, t \cdot t, 1\right) \cdot x \]

    2. lift-fma.f64 N/A

       \[\leadsto \left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot \left(t \cdot t\right) + 1\right) \cdot x \]

    3. associate-*r* N/A

       \[\leadsto \left(\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t\right) \cdot t + 1\right) \cdot x \]

    4. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t, t, 1\right) \cdot x \]

    5. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t, t, 1\right) \cdot x \]

    6. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t, t, 1\right) \cdot x \]

    7. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\left({b}^{2} \cdot \frac{-1}{512}\right) \cdot t, t, 1\right) \cdot x \]

    8. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{512} \cdot {b}^{2}\right) \cdot t, t, 1\right) \cdot x \]

    9. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{512} \cdot {b}^{2}\right) \cdot t, t, 1\right) \cdot x \]

    10. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(\left({b}^{2} \cdot \frac{-1}{512}\right) \cdot t, t, 1\right) \cdot x \]

    11. pow2 N/A

        \[\leadsto \mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t, t, 1\right) \cdot x \]

    12. lift-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t, t, 1\right) \cdot x \]

    13. lift-*.f64 25.6

        \[\leadsto \mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot -0.001953125\right) \cdot t, t, 1\right) \cdot x \]

12. Applied rewrites 25.6%

    \[\leadsto \mathsf{fma}\left(\left(\left(b \cdot b\right) \cdot -0.001953125\right) \cdot t, t, 1\right) \cdot x \]
13. Add Preprocessing

Alternative 4: 24.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(b \cdot b\right) \cdot -0.001953125, t \cdot t, 1\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* (fma (* (* b b) -0.001953125) (* t t) 1.0) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(((b * b) * -0.001953125), (t * t), 1.0) * x;
}

Julia

function code(x, y, z, t, a, b)
	return Float64(fma(Float64(Float64(b * b) * -0.001953125), Float64(t * t), 1.0) * x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(b * b), $MachinePrecision] * -0.001953125), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 27.7%

   \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \color{blue}{x} \]

   2. lower-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \color{blue}{x} \]

   3. lower-cos.f64 N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x \]

   4. *-commutative N/A

      \[\leadsto \cos \left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right) \cdot x \]

   5. lower-*.f64 N/A

      \[\leadsto \cos \left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right) \cdot x \]

   6. *-commutative N/A

      \[\leadsto \cos \left(\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   7. lower-*.f64 N/A

      \[\leadsto \cos \left(\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   8. *-commutative N/A

      \[\leadsto \cos \left(\left(\left(\left(1 + 2 \cdot a\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   9. lower-*.f64 N/A

      \[\leadsto \cos \left(\left(\left(\left(1 + 2 \cdot a\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   10. +-commutative N/A

       \[\leadsto \cos \left(\left(\left(\left(2 \cdot a + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   11. *-commutative N/A

       \[\leadsto \cos \left(\left(\left(\left(a \cdot 2 + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   12. lower-fma.f64 29.0

       \[\leadsto \cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x \]

4. Applied rewrites 29.0%

   \[\leadsto \color{blue}{\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x} \]

5. Taylor expanded in a around 0

   \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]
6. Step-by-step derivation

   1. lower-cos.f64 N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]

   2. *-commutative N/A

      \[\leadsto \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \cdot x \]

   3. lift-*.f64 N/A

      \[\leadsto \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \cdot x \]

   4. lift-*.f64 29.9

      \[\leadsto \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot x \]

7. Applied rewrites 29.9%

   \[\leadsto \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot x \]

8. Taylor expanded in t around 0

   \[\leadsto \left(1 + \frac{-1}{512} \cdot \left({b}^{2} \cdot {t}^{2}\right)\right) \cdot x \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {t}^{2}\right) + 1\right) \cdot x \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-1}{512} \cdot {b}^{2}\right) \cdot {t}^{2} + 1\right) \cdot x \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{512} \cdot {b}^{2}, {t}^{2}, 1\right) \cdot x \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

   6. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

   7. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

   8. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{-1}{512}, t \cdot t, 1\right) \cdot x \]

   9. lift-*.f64 24.7

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot -0.001953125, t \cdot t, 1\right) \cdot x \]

10. Applied rewrites 24.7%

    \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot -0.001953125, t \cdot t, 1\right) \cdot x \]
11. Add Preprocessing

Alternative 5: 3.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(b \cdot b\right) \cdot -0.001953125\right) \cdot t\right) \cdot t\right) \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* (* (* (* (* b b) -0.001953125) t) t) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((((b * b) * -0.001953125) * t) * t) * x;
}

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(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((b * b) * (-0.001953125d0)) * t) * t) * x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((((b * b) * -0.001953125) * t) * t) * x;
}

Python

def code(x, y, z, t, a, b):
	return ((((b * b) * -0.001953125) * t) * t) * x

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(Float64(Float64(b * b) * -0.001953125) * t) * t) * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((((b * b) * -0.001953125) * t) * t) * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(N[(b * b), $MachinePrecision] * -0.001953125), $MachinePrecision] * t), $MachinePrecision] * t), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 27.7%

   \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x \cdot \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \color{blue}{x} \]

   2. lower-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot \color{blue}{x} \]

   3. lower-cos.f64 N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right)\right) \cdot x \]

   4. *-commutative N/A

      \[\leadsto \cos \left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right) \cdot x \]

   5. lower-*.f64 N/A

      \[\leadsto \cos \left(\left(b \cdot \left(t \cdot \left(1 + 2 \cdot a\right)\right)\right) \cdot \frac{1}{16}\right) \cdot x \]

   6. *-commutative N/A

      \[\leadsto \cos \left(\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   7. lower-*.f64 N/A

      \[\leadsto \cos \left(\left(\left(t \cdot \left(1 + 2 \cdot a\right)\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   8. *-commutative N/A

      \[\leadsto \cos \left(\left(\left(\left(1 + 2 \cdot a\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   9. lower-*.f64 N/A

      \[\leadsto \cos \left(\left(\left(\left(1 + 2 \cdot a\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   10. +-commutative N/A

       \[\leadsto \cos \left(\left(\left(\left(2 \cdot a + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   11. *-commutative N/A

       \[\leadsto \cos \left(\left(\left(\left(a \cdot 2 + 1\right) \cdot t\right) \cdot b\right) \cdot \frac{1}{16}\right) \cdot x \]

   12. lower-fma.f64 29.0

       \[\leadsto \cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x \]

4. Applied rewrites 29.0%

   \[\leadsto \color{blue}{\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x} \]

5. Taylor expanded in a around 0

   \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]
6. Step-by-step derivation

   1. lower-cos.f64 N/A

      \[\leadsto \cos \left(\frac{1}{16} \cdot \left(b \cdot t\right)\right) \cdot x \]

   2. *-commutative N/A

      \[\leadsto \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \cdot x \]

   3. lift-*.f64 N/A

      \[\leadsto \cos \left(\left(b \cdot t\right) \cdot \frac{1}{16}\right) \cdot x \]

   4. lift-*.f64 29.9

      \[\leadsto \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot x \]

7. Applied rewrites 29.9%

   \[\leadsto \cos \left(\left(b \cdot t\right) \cdot 0.0625\right) \cdot x \]

8. Taylor expanded in t around 0

   \[\leadsto \left(1 + \frac{-1}{512} \cdot \left({b}^{2} \cdot {t}^{2}\right)\right) \cdot x \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {t}^{2}\right) + 1\right) \cdot x \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(\frac{-1}{512} \cdot {b}^{2}\right) \cdot {t}^{2} + 1\right) \cdot x \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{512} \cdot {b}^{2}, {t}^{2}, 1\right) \cdot x \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

   6. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

   7. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{-1}{512}, {t}^{2}, 1\right) \cdot x \]

   8. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot \frac{-1}{512}, t \cdot t, 1\right) \cdot x \]

   9. lift-*.f64 24.7

      \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot -0.001953125, t \cdot t, 1\right) \cdot x \]

10. Applied rewrites 24.7%

    \[\leadsto \mathsf{fma}\left(\left(b \cdot b\right) \cdot -0.001953125, t \cdot t, 1\right) \cdot x \]

11. Taylor expanded in t around inf

    \[\leadsto \left(\frac{-1}{512} \cdot \left({b}^{2} \cdot {t}^{2}\right)\right) \cdot x \]
12. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \left(\left(\frac{-1}{512} \cdot {b}^{2}\right) \cdot {t}^{2}\right) \cdot x \]

    2. *-commutative N/A

       \[\leadsto \left(\left({b}^{2} \cdot \frac{-1}{512}\right) \cdot {t}^{2}\right) \cdot x \]

    3. pow2 N/A

       \[\leadsto \left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot {t}^{2}\right) \cdot x \]

    4. lift-*.f64 N/A

       \[\leadsto \left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot {t}^{2}\right) \cdot x \]

    5. lift-*.f64 N/A

       \[\leadsto \left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot {t}^{2}\right) \cdot x \]

    6. pow2 N/A

       \[\leadsto \left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot \left(t \cdot t\right)\right) \cdot x \]

    7. associate-*r* N/A

       \[\leadsto \left(\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t\right) \cdot t\right) \cdot x \]

    8. lower-*.f64 N/A

       \[\leadsto \left(\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t\right) \cdot t\right) \cdot x \]

    9. lift-*.f64 N/A

       \[\leadsto \left(\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t\right) \cdot t\right) \cdot x \]

    10. lift-*.f64 N/A

        \[\leadsto \left(\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t\right) \cdot t\right) \cdot x \]

    11. pow2 N/A

        \[\leadsto \left(\left(\left({b}^{2} \cdot \frac{-1}{512}\right) \cdot t\right) \cdot t\right) \cdot x \]

    12. *-commutative N/A

        \[\leadsto \left(\left(\left(\frac{-1}{512} \cdot {b}^{2}\right) \cdot t\right) \cdot t\right) \cdot x \]

    13. lower-*.f64 N/A

        \[\leadsto \left(\left(\left(\frac{-1}{512} \cdot {b}^{2}\right) \cdot t\right) \cdot t\right) \cdot x \]

    14. *-commutative N/A

        \[\leadsto \left(\left(\left({b}^{2} \cdot \frac{-1}{512}\right) \cdot t\right) \cdot t\right) \cdot x \]

    15. pow2 N/A

        \[\leadsto \left(\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t\right) \cdot t\right) \cdot x \]

    16. lift-*.f64 N/A

        \[\leadsto \left(\left(\left(\left(b \cdot b\right) \cdot \frac{-1}{512}\right) \cdot t\right) \cdot t\right) \cdot x \]

    17. lift-*.f64 3.2

        \[\leadsto \left(\left(\left(\left(b \cdot b\right) \cdot -0.001953125\right) \cdot t\right) \cdot t\right) \cdot x \]

13. Applied rewrites 3.2%

    \[\leadsto \left(\left(\left(\left(b \cdot b\right) \cdot -0.001953125\right) \cdot t\right) \cdot t\right) \cdot x \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025139 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64
  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))