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

Percentage Accurate: 27.5% → 31.7%

Time: 21.9s

Alternatives: 5

Speedup: 225.0×

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

real(8) function code(x, y, z, t, a, b)
    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: 27.5% 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

real(8) function code(x, y, z, t, a, b)
    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: 31.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t_1}{16}\right) \leq 2 \cdot 10^{+224}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{z \cdot \mathsf{fma}\left(y, 2, 1\right)}{\frac{16}{t}}\right) \cdot \cos \left(0.0625 \cdot t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(e^{\log 2 + -0.0009765625 \cdot \left(t \cdot \left(t \cdot \left(z \cdot z\right)\right)\right)} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* t (* b (+ 1.0 (* 2.0 a))))))
   (if (<=
        (*
         (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
         (cos (/ t_1 16.0)))
        2e+224)
     (* x (* (cos (/ (* z (fma y 2.0 1.0)) (/ 16.0 t))) (cos (* 0.0625 t_1))))
     (*
      x
      (+ (exp (+ (log 2.0) (* -0.0009765625 (* t (* t (* z z)))))) -1.0)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t * (b * (1.0 + (2.0 * a)));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((t_1 / 16.0))) <= 2e+224) {
		tmp = x * (cos(((z * fma(y, 2.0, 1.0)) / (16.0 / t))) * cos((0.0625 * t_1)));
	} else {
		tmp = x * (exp((log(2.0) + (-0.0009765625 * (t * (t * (z * z)))))) + -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t * Float64(b * Float64(1.0 + Float64(2.0 * a))))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(t_1 / 16.0))) <= 2e+224)
		tmp = Float64(x * Float64(cos(Float64(Float64(z * fma(y, 2.0, 1.0)) / Float64(16.0 / t))) * cos(Float64(0.0625 * t_1))));
	else
		tmp = Float64(x * Float64(exp(Float64(log(2.0) + Float64(-0.0009765625 * Float64(t * Float64(t * Float64(z * z)))))) + -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t * N[(b * N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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[(t$95$1 / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+224], N[(x * N[(N[Cos[N[(N[(z * N[(y * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(16.0 / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.0625 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] + N[(-0.0009765625 * N[(t * N[(t * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 1.99999999999999994e224

   1. Initial program 46.6%

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

      1. associate-*l* 46.6%

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

      2. associate-/l* 46.6%

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

      3. fma-def 46.6%

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

      4. associate-/l* 46.0%

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

      5. fma-def 46.0%

         \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \left(\frac{\color{blue}{\mathsf{fma}\left(a, 2, 1\right)} \cdot b}{\frac{16}{t}}\right)\right) \]

   3. Simplified 46.0%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right)} \]

   4. Taylor expanded in b around 0 46.6%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \color{blue}{\left(0.0625 \cdot \left(t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)}\right) \]

   if 1.99999999999999994e224 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 3.4%

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

      1. associate-*l* 3.4%

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

   3. Simplified 5.0%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + \frac{a}{8}\right)\right)\right)} \]

   4. Taylor expanded in t around 0 9.1%

      \[\leadsto x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \color{blue}{1}\right) \]

   5. Taylor expanded in y around 0 10.0%

      \[\leadsto \color{blue}{\cos \left(0.0625 \cdot \left(t \cdot z\right)\right) \cdot x} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 10.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right)\right)} \cdot x \]

      2. expm1-udef 10.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right)} - 1\right)} \cdot x \]

      3. *-commutative 10.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\left(t \cdot z\right) \cdot 0.0625\right)}\right)} - 1\right) \cdot x \]

   7. Applied egg-rr 10.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)} - 1\right)} \cdot x \]

   8. Taylor expanded in t around 0 12.1%

      \[\leadsto \left(e^{\color{blue}{\log 2 + -0.0009765625 \cdot \left({t}^{2} \cdot {z}^{2}\right)}} - 1\right) \cdot x \]
   9. Step-by-step derivation

      1. unpow2 12.1%

         \[\leadsto \left(e^{\log 2 + -0.0009765625 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot {z}^{2}\right)} - 1\right) \cdot x \]

      2. associate-*l* 14.0%

         \[\leadsto \left(e^{\log 2 + -0.0009765625 \cdot \color{blue}{\left(t \cdot \left(t \cdot {z}^{2}\right)\right)}} - 1\right) \cdot x \]

      3. unpow2 14.0%

         \[\leadsto \left(e^{\log 2 + -0.0009765625 \cdot \left(t \cdot \left(t \cdot \color{blue}{\left(z \cdot z\right)}\right)\right)} - 1\right) \cdot x \]

   10. Simplified 14.0%

       \[\leadsto \left(e^{\color{blue}{\log 2 + -0.0009765625 \cdot \left(t \cdot \left(t \cdot \left(z \cdot z\right)\right)\right)}} - 1\right) \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 2 \cdot 10^{+224}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{z \cdot \mathsf{fma}\left(y, 2, 1\right)}{\frac{16}{t}}\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(e^{\log 2 + -0.0009765625 \cdot \left(t \cdot \left(t \cdot \left(z \cdot z\right)\right)\right)} + -1\right)\\ \end{array} \]

Alternative 2: 31.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right)\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+224}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(e^{\log 2 + -0.0009765625 \cdot \left(t \cdot \left(t \cdot \left(z \cdot z\right)\right)\right)} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (*
          (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
          (cos (/ (* t (* b (+ 1.0 (* 2.0 a)))) 16.0)))))
   (if (<= t_1 2e+224)
     t_1
     (*
      x
      (+ (exp (+ (log 2.0) (* -0.0009765625 (* t (* t (* z z)))))) -1.0)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * (b * (1.0 + (2.0 * a)))) / 16.0));
	double tmp;
	if (t_1 <= 2e+224) {
		tmp = t_1;
	} else {
		tmp = x * (exp((log(2.0) + (-0.0009765625 * (t * (t * (z * z)))))) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos(((t * (b * (1.0d0 + (2.0d0 * a)))) / 16.0d0))
    if (t_1 <= 2d+224) then
        tmp = t_1
    else
        tmp = x * (exp((log(2.0d0) + ((-0.0009765625d0) * (t * (t * (z * z)))))) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos(((t * (b * (1.0 + (2.0 * a)))) / 16.0));
	double tmp;
	if (t_1 <= 2e+224) {
		tmp = t_1;
	} else {
		tmp = x * (Math.exp((Math.log(2.0) + (-0.0009765625 * (t * (t * (z * z)))))) + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos(((t * (b * (1.0 + (2.0 * a)))) / 16.0))
	tmp = 0
	if t_1 <= 2e+224:
		tmp = t_1
	else:
		tmp = x * (math.exp((math.log(2.0) + (-0.0009765625 * (t * (t * (z * z)))))) + -1.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(b * Float64(1.0 + Float64(2.0 * a)))) / 16.0)))
	tmp = 0.0
	if (t_1 <= 2e+224)
		tmp = t_1;
	else
		tmp = Float64(x * Float64(exp(Float64(log(2.0) + Float64(-0.0009765625 * Float64(t * Float64(t * Float64(z * z)))))) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * (b * (1.0 + (2.0 * a)))) / 16.0));
	tmp = 0.0;
	if (t_1 <= 2e+224)
		tmp = t_1;
	else
		tmp = x * (exp((log(2.0) + (-0.0009765625 * (t * (t * (z * z)))))) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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[(t * N[(b * N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+224], t$95$1, N[(x * N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] + N[(-0.0009765625 * N[(t * N[(t * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 1.99999999999999994e224

   1. Initial program 46.6%

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

   if 1.99999999999999994e224 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 3.4%

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

      1. associate-*l* 3.4%

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

   3. Simplified 5.0%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + \frac{a}{8}\right)\right)\right)} \]

   4. Taylor expanded in t around 0 9.1%

      \[\leadsto x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \color{blue}{1}\right) \]

   5. Taylor expanded in y around 0 10.0%

      \[\leadsto \color{blue}{\cos \left(0.0625 \cdot \left(t \cdot z\right)\right) \cdot x} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 10.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right)\right)} \cdot x \]

      2. expm1-udef 10.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right)} - 1\right)} \cdot x \]

      3. *-commutative 10.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\left(t \cdot z\right) \cdot 0.0625\right)}\right)} - 1\right) \cdot x \]

   7. Applied egg-rr 10.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)} - 1\right)} \cdot x \]

   8. Taylor expanded in t around 0 12.1%

      \[\leadsto \left(e^{\color{blue}{\log 2 + -0.0009765625 \cdot \left({t}^{2} \cdot {z}^{2}\right)}} - 1\right) \cdot x \]
   9. Step-by-step derivation

      1. unpow2 12.1%

         \[\leadsto \left(e^{\log 2 + -0.0009765625 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot {z}^{2}\right)} - 1\right) \cdot x \]

      2. associate-*l* 14.0%

         \[\leadsto \left(e^{\log 2 + -0.0009765625 \cdot \color{blue}{\left(t \cdot \left(t \cdot {z}^{2}\right)\right)}} - 1\right) \cdot x \]

      3. unpow2 14.0%

         \[\leadsto \left(e^{\log 2 + -0.0009765625 \cdot \left(t \cdot \left(t \cdot \color{blue}{\left(z \cdot z\right)}\right)\right)} - 1\right) \cdot x \]

   10. Simplified 14.0%

       \[\leadsto \left(e^{\color{blue}{\log 2 + -0.0009765625 \cdot \left(t \cdot \left(t \cdot \left(z \cdot z\right)\right)\right)}} - 1\right) \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right) \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(b \cdot \left(1 + 2 \cdot a\right)\right)}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(e^{\log 2 + -0.0009765625 \cdot \left(t \cdot \left(t \cdot \left(z \cdot z\right)\right)\right)} + -1\right)\\ \end{array} \]

Alternative 3: 29.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.16 \cdot 10^{-38}:\\ \;\;\;\;x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + \frac{a}{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(e^{\log 2 + -0.0009765625 \cdot \left(t \cdot \left(t \cdot \left(z \cdot z\right)\right)\right)} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 1.16e-38)
   (*
    x
    (*
     (cos (* (* z t) (+ 0.0625 (/ y 8.0))))
     (cos (* (* t b) (+ 0.0625 (/ a 8.0))))))
   (* x (+ (exp (+ (log 2.0) (* -0.0009765625 (* t (* t (* z z)))))) -1.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 1.16e-38) {
		tmp = x * (cos(((z * t) * (0.0625 + (y / 8.0)))) * cos(((t * b) * (0.0625 + (a / 8.0)))));
	} else {
		tmp = x * (exp((log(2.0) + (-0.0009765625 * (t * (t * (z * z)))))) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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
    real(8) :: tmp
    if (t <= 1.16d-38) then
        tmp = x * (cos(((z * t) * (0.0625d0 + (y / 8.0d0)))) * cos(((t * b) * (0.0625d0 + (a / 8.0d0)))))
    else
        tmp = x * (exp((log(2.0d0) + ((-0.0009765625d0) * (t * (t * (z * z)))))) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 1.16e-38) {
		tmp = x * (Math.cos(((z * t) * (0.0625 + (y / 8.0)))) * Math.cos(((t * b) * (0.0625 + (a / 8.0)))));
	} else {
		tmp = x * (Math.exp((Math.log(2.0) + (-0.0009765625 * (t * (t * (z * z)))))) + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 1.16e-38:
		tmp = x * (math.cos(((z * t) * (0.0625 + (y / 8.0)))) * math.cos(((t * b) * (0.0625 + (a / 8.0)))))
	else:
		tmp = x * (math.exp((math.log(2.0) + (-0.0009765625 * (t * (t * (z * z)))))) + -1.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 1.16e-38)
		tmp = Float64(x * Float64(cos(Float64(Float64(z * t) * Float64(0.0625 + Float64(y / 8.0)))) * cos(Float64(Float64(t * b) * Float64(0.0625 + Float64(a / 8.0))))));
	else
		tmp = Float64(x * Float64(exp(Float64(log(2.0) + Float64(-0.0009765625 * Float64(t * Float64(t * Float64(z * z)))))) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 1.16e-38)
		tmp = x * (cos(((z * t) * (0.0625 + (y / 8.0)))) * cos(((t * b) * (0.0625 + (a / 8.0)))));
	else
		tmp = x * (exp((log(2.0) + (-0.0009765625 * (t * (t * (z * z)))))) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 1.16e-38], N[(x * N[(N[Cos[N[(N[(z * t), $MachinePrecision] * N[(0.0625 + N[(y / 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t * b), $MachinePrecision] * N[(0.0625 + N[(a / 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] + N[(-0.0009765625 * N[(t * N[(t * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.15999999999999995e-38

   1. Initial program 34.4%

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

      1. associate-*l* 34.4%

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

   3. Simplified 35.3%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + \frac{a}{8}\right)\right)\right)} \]

   if 1.15999999999999995e-38 < t

   1. Initial program 7.8%

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

      1. associate-*l* 7.8%

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

   3. Simplified 7.6%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + \frac{a}{8}\right)\right)\right)} \]

   4. Taylor expanded in t around 0 10.2%

      \[\leadsto x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \color{blue}{1}\right) \]

   5. Taylor expanded in y around 0 11.3%

      \[\leadsto \color{blue}{\cos \left(0.0625 \cdot \left(t \cdot z\right)\right) \cdot x} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 11.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right)\right)} \cdot x \]

      2. expm1-udef 11.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(0.0625 \cdot \left(t \cdot z\right)\right)\right)} - 1\right)} \cdot x \]

      3. *-commutative 11.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\cos \color{blue}{\left(\left(t \cdot z\right) \cdot 0.0625\right)}\right)} - 1\right) \cdot x \]

   7. Applied egg-rr 11.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\left(t \cdot z\right) \cdot 0.0625\right)\right)} - 1\right)} \cdot x \]

   8. Taylor expanded in t around 0 14.4%

      \[\leadsto \left(e^{\color{blue}{\log 2 + -0.0009765625 \cdot \left({t}^{2} \cdot {z}^{2}\right)}} - 1\right) \cdot x \]
   9. Step-by-step derivation

      1. unpow2 14.4%

         \[\leadsto \left(e^{\log 2 + -0.0009765625 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot {z}^{2}\right)} - 1\right) \cdot x \]

      2. associate-*l* 15.2%

         \[\leadsto \left(e^{\log 2 + -0.0009765625 \cdot \color{blue}{\left(t \cdot \left(t \cdot {z}^{2}\right)\right)}} - 1\right) \cdot x \]

      3. unpow2 15.2%

         \[\leadsto \left(e^{\log 2 + -0.0009765625 \cdot \left(t \cdot \left(t \cdot \color{blue}{\left(z \cdot z\right)}\right)\right)} - 1\right) \cdot x \]

   10. Simplified 15.2%

       \[\leadsto \left(e^{\color{blue}{\log 2 + -0.0009765625 \cdot \left(t \cdot \left(t \cdot \left(z \cdot z\right)\right)\right)}} - 1\right) \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.16 \cdot 10^{-38}:\\ \;\;\;\;x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + \frac{a}{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(e^{\log 2 + -0.0009765625 \cdot \left(t \cdot \left(t \cdot \left(z \cdot z\right)\right)\right)} + -1\right)\\ \end{array} \]

Alternative 4: 29.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{+69}:\\ \;\;\;\;x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(t \cdot \left(b \cdot 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 9.5e+69)
   (* x (* (cos (* (* z t) (+ 0.0625 (/ y 8.0)))) (cos (* t (* b 0.0625)))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 9.5e+69) {
		tmp = x * (cos(((z * t) * (0.0625 + (y / 8.0)))) * cos((t * (b * 0.0625))));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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
    real(8) :: tmp
    if (t <= 9.5d+69) then
        tmp = x * (cos(((z * t) * (0.0625d0 + (y / 8.0d0)))) * cos((t * (b * 0.0625d0))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 9.5e+69) {
		tmp = x * (Math.cos(((z * t) * (0.0625 + (y / 8.0)))) * Math.cos((t * (b * 0.0625))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 9.5e+69:
		tmp = x * (math.cos(((z * t) * (0.0625 + (y / 8.0)))) * math.cos((t * (b * 0.0625))))
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 9.5e+69)
		tmp = Float64(x * Float64(cos(Float64(Float64(z * t) * Float64(0.0625 + Float64(y / 8.0)))) * cos(Float64(t * Float64(b * 0.0625)))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 9.5e+69)
		tmp = x * (cos(((z * t) * (0.0625 + (y / 8.0)))) * cos((t * (b * 0.0625))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 9.5e+69], N[(x * N[(N[Cos[N[(N[(z * t), $MachinePrecision] * N[(0.0625 + N[(y / 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(t * N[(b * 0.0625), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 9.4999999999999995e69

   1. Initial program 32.1%

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

      1. associate-*l* 32.1%

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

   3. Simplified 32.8%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + \frac{a}{8}\right)\right)\right)} \]

   4. Taylor expanded in a around 0 33.1%

      \[\leadsto x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \color{blue}{\cos \left(0.0625 \cdot \left(t \cdot b\right)\right)}\right) \]
   5. Step-by-step derivation

      1. *-commutative 33.1%

         \[\leadsto x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \color{blue}{\left(\left(t \cdot b\right) \cdot 0.0625\right)}\right) \]

      2. associate-*l* 33.1%

         \[\leadsto x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \color{blue}{\left(t \cdot \left(b \cdot 0.0625\right)\right)}\right) \]

   6. Simplified 33.1%

      \[\leadsto x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \color{blue}{\cos \left(t \cdot \left(b \cdot 0.0625\right)\right)}\right) \]

   if 9.4999999999999995e69 < t

   1. Initial program 5.3%

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

      1. associate-*l* 5.3%

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

   3. Simplified 5.3%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + \frac{a}{8}\right)\right)\right)} \]

   4. Taylor expanded in t around 0 6.8%

      \[\leadsto x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \color{blue}{1}\right) \]

   5. Taylor expanded in z around 0 13.5%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{+69}:\\ \;\;\;\;x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(t \cdot \left(b \cdot 0.0625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 30.6% accurate, 225.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    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
end function

Java

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

Python

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

Julia

function code(x, y, z, t, a, b)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 27.0%

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

   1. associate-*l* 27.0%

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

3. Simplified 27.6%

   \[\leadsto \color{blue}{x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + \frac{a}{8}\right)\right)\right)} \]

4. Taylor expanded in t around 0 28.2%

   \[\leadsto x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \color{blue}{1}\right) \]

5. Taylor expanded in z around 0 29.1%

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

6. Final simplification 29.1%

   \[\leadsto x \]

Developer target: 30.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    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(((b / 16.0d0) * (t / ((1.0d0 - (a * 2.0d0)) + ((a * 2.0d0) ** 2.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023199 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))