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

Percentage Accurate: 27.0% → 31.5%

Time: 14.9s

Alternatives: 4

Speedup: 13.5×

Specification

Math

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

FPCore

(FPCore (x y z t a b)
  :precision binary64
  :pre TRUE
  (*
 (* 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)
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]

ReFlow

f(x, y, z, t, a, b):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf],
	t in [-inf, +inf],
	a in [-inf, +inf],
	b in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z, t, a, b: real): real =
	(x * (cos((((((y * (2)) + (1)) * z) * t) / (16))))) * (cos((((((a * (2)) + (1)) * b) * t) / (16))))
END code

Initial Program: 27.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b)
  :precision binary64
  :pre TRUE
  (*
 (* 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)
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]

ReFlow

f(x, y, z, t, a, b):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf],
	t in [-inf, +inf],
	a in [-inf, +inf],
	b in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z, t, a, b: real): real =
	(x * (cos((((((y * (2)) + (1)) * z) * t) / (16))))) * (cos((((((a * (2)) + (1)) * b) * t) / (16))))
END code

Alternative 1: 31.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot \left|t\right|}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \left|t\right|}{16}\right) \leq 10^{+288}:\\ \;\;\;\;\left(x \cdot \sin \left(\mathsf{fma}\left(\left|t\right| \cdot 0.0625, z \cdot \mathsf{fma}\left(2, y, 1\right), \pi \cdot 0.5\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot \left|t\right|, 0.0625, \pi \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{2}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  :pre TRUE
  (if (<=
     (*
      (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) (fabs t)) 16.0)))
      (cos (/ (* (* (+ (* a 2.0) 1.0) b) (fabs t)) 16.0)))
     1e+288)
  (*
   (*
    x
    (sin (fma (* (fabs t) 0.0625) (* z (fma 2.0 y 1.0)) (* PI 0.5))))
   (sin (fma (* (* (fma a 2.0 1.0) b) (fabs t)) 0.0625 (* PI 0.5))))
  (* x (/ 2.0 2.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * fabs(t)) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * fabs(t)) / 16.0))) <= 1e+288) {
		tmp = (x * sin(fma((fabs(t) * 0.0625), (z * fma(2.0, y, 1.0)), (((double) M_PI) * 0.5)))) * sin(fma(((fma(a, 2.0, 1.0) * b) * fabs(t)), 0.0625, (((double) M_PI) * 0.5)));
	} else {
		tmp = x * (2.0 / 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * abs(t)) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * abs(t)) / 16.0))) <= 1e+288)
		tmp = Float64(Float64(x * sin(fma(Float64(abs(t) * 0.0625), Float64(z * fma(2.0, y, 1.0)), Float64(pi * 0.5)))) * sin(fma(Float64(Float64(fma(a, 2.0, 1.0) * b) * abs(t)), 0.0625, Float64(pi * 0.5))));
	else
		tmp = Float64(x * Float64(2.0 / 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+288], N[(N[(x * N[Sin[N[(N[(N[Abs[t], $MachinePrecision] * 0.0625), $MachinePrecision] * N[(z * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * 0.0625 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / 2.0), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x, y, z, t, a, b):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf],
	t in [-inf, +inf],
	a in [-inf, +inf],
	b in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z, t, a, b: real): real =
	LET tmp = IF (((x * (cos((((((y * (2)) + (1)) * z) * (abs(t))) / (16))))) * (cos((((((a * (2)) + (1)) * b) * (abs(t))) / (16))))) <= (1000000000000000007630473539575035660514778335511710750780086664439969510636494954611131549135839186513983455555395220895687860544809584999829725260594873271087399626486606146442550988840016917394626449536395208620267012778077787723395914064607119962069483324573977857832138825282954985472)) THEN ((x * (sin(((((abs(t)) * (625e-4)) * (z * (((2) * y) + (1)))) + ((4 * atan(1)) * (5e-1)))))) * (sin(((((((a * (2)) + (1)) * b) * (abs(t))) * (625e-4)) + ((4 * atan(1)) * (5e-1)))))) ELSE (x * ((2) / (2))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1e288

   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

   3. Applied rewrites 26.9%

      \[\leadsto \left(x \cdot \sin \left(\mathsf{fma}\left(t \cdot 0.0625, z \cdot \mathsf{fma}\left(2, y, 1\right), \pi \cdot 0.5\right)\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 26.9%

      \[\leadsto \left(x \cdot \sin \left(\mathsf{fma}\left(t \cdot 0.0625, z \cdot \mathsf{fma}\left(2, y, 1\right), \pi \cdot 0.5\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot t, 0.0625, \pi \cdot 0.5\right)\right) \]

   if 1e288 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   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. Taylor expanded in a around 0

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

   4. Applied rewrites 27.7%

      \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   6. Applied rewrites 26.7%

      \[\leadsto x \cdot \frac{\sin \left(\mathsf{fma}\left(\pi, 0.5, \left(\left(0.0625 \cdot t\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot z\right) - \left(b \cdot t\right) \cdot 0.0625\right) + \sin \left(\mathsf{fma}\left(\pi, 0.5, \left(\left(0.0625 \cdot t\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot z\right) + \left(b \cdot t\right) \cdot 0.0625\right)}{2} \]

   7. Taylor expanded in t around 0

      \[\leadsto x \cdot \frac{2}{2} \]
   8. Step-by-step derivation

   9. Applied rewrites 30.3%

      \[\leadsto x \cdot \frac{2}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 31.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot \left|t\right|}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot \left|t\right|}{16}\right) \leq 10^{+288}:\\ \;\;\;\;\left(x \cdot \sin \left(\mathsf{fma}\left(0.0625, \left|t\right| \cdot z, 0.5 \cdot \pi\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot \left|t\right|, 0.0625, 1.5707963267948966\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{2}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  :pre TRUE
  (if (<=
     (*
      (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) (fabs t)) 16.0)))
      (cos (/ (* (* (+ (* a 2.0) 1.0) b) (fabs t)) 16.0)))
     1e+288)
  (*
   (* x (sin (fma 0.0625 (* (fabs t) z) (* 0.5 PI))))
   (sin
    (fma
     (* (* (fma a 2.0 1.0) b) (fabs t))
     0.0625
     1.5707963267948966)))
  (* x (/ 2.0 2.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * fabs(t)) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * fabs(t)) / 16.0))) <= 1e+288) {
		tmp = (x * sin(fma(0.0625, (fabs(t) * z), (0.5 * ((double) M_PI))))) * sin(fma(((fma(a, 2.0, 1.0) * b) * fabs(t)), 0.0625, 1.5707963267948966));
	} else {
		tmp = x * (2.0 / 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * abs(t)) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * abs(t)) / 16.0))) <= 1e+288)
		tmp = Float64(Float64(x * sin(fma(0.0625, Float64(abs(t) * z), Float64(0.5 * pi)))) * sin(fma(Float64(Float64(fma(a, 2.0, 1.0) * b) * abs(t)), 0.0625, 1.5707963267948966)));
	else
		tmp = Float64(x * Float64(2.0 / 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+288], N[(N[(x * N[Sin[N[(0.0625 * N[(N[Abs[t], $MachinePrecision] * z), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * b), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * 0.0625 + 1.5707963267948966), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / 2.0), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x, y, z, t, a, b):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf],
	t in [-inf, +inf],
	a in [-inf, +inf],
	b in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z, t, a, b: real): real =
	LET tmp = IF (((x * (cos((((((y * (2)) + (1)) * z) * (abs(t))) / (16))))) * (cos((((((a * (2)) + (1)) * b) * (abs(t))) / (16))))) <= (1000000000000000007630473539575035660514778335511710750780086664439969510636494954611131549135839186513983455555395220895687860544809584999829725260594873271087399626486606146442550988840016917394626449536395208620267012778077787723395914064607119962069483324573977857832138825282954985472)) THEN ((x * (sin((((625e-4) * ((abs(t)) * z)) + ((5e-1) * (4 * atan(1))))))) * (sin(((((((a * (2)) + (1)) * b) * (abs(t))) * (625e-4)) + (15707963267948965579989817342720925807952880859375e-49))))) ELSE (x * ((2) / (2))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 1e288

   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

   3. Applied rewrites 26.9%

      \[\leadsto \left(x \cdot \sin \left(\mathsf{fma}\left(t \cdot 0.0625, z \cdot \mathsf{fma}\left(2, y, 1\right), \pi \cdot 0.5\right)\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 26.9%

      \[\leadsto \left(x \cdot \sin \left(\mathsf{fma}\left(t \cdot 0.0625, z \cdot \mathsf{fma}\left(2, y, 1\right), \pi \cdot 0.5\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot t, 0.0625, \pi \cdot 0.5\right)\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(x \cdot \sin \left(\frac{1}{16} \cdot \left(t \cdot z\right) + \frac{1}{2} \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot t, 0.0625, \pi \cdot 0.5\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 27.7%

      \[\leadsto \left(x \cdot \sin \left(\mathsf{fma}\left(0.0625, t \cdot z, 0.5 \cdot \pi\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot t, 0.0625, \pi \cdot 0.5\right)\right) \]

   9. Evaluated real constant 27.7%

      \[\leadsto \left(x \cdot \sin \left(\mathsf{fma}\left(0.0625, t \cdot z, 0.5 \cdot \pi\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot t, 0.0625, 1.5707963267948966\right)\right) \]

   if 1e288 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   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. Taylor expanded in a around 0

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

   4. Applied rewrites 27.7%

      \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   6. Applied rewrites 26.7%

      \[\leadsto x \cdot \frac{\sin \left(\mathsf{fma}\left(\pi, 0.5, \left(\left(0.0625 \cdot t\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot z\right) - \left(b \cdot t\right) \cdot 0.0625\right) + \sin \left(\mathsf{fma}\left(\pi, 0.5, \left(\left(0.0625 \cdot t\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot z\right) + \left(b \cdot t\right) \cdot 0.0625\right)}{2} \]

   7. Taylor expanded in t around 0

      \[\leadsto x \cdot \frac{2}{2} \]
   8. Step-by-step derivation

   9. Applied rewrites 30.3%

      \[\leadsto x \cdot \frac{2}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 30.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|t\right| \leq 2.864484426612483 \cdot 10^{-41}:\\ \;\;\;\;x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot \left|t\right|\right)\right) \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \left(\left(0.0625 \cdot \left|t\right|\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{2}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  :pre TRUE
  (if (<= (fabs t) 2.864484426612483e-41)
  (*
   x
   (*
    (cos (* 0.0625 (* b (fabs t))))
    (sin (fma PI 0.5 (* (* (* 0.0625 (fabs t)) (fma 2.0 y 1.0)) z)))))
  (* x (/ 2.0 2.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (fabs(t) <= 2.864484426612483e-41) {
		tmp = x * (cos((0.0625 * (b * fabs(t)))) * sin(fma(((double) M_PI), 0.5, (((0.0625 * fabs(t)) * fma(2.0, y, 1.0)) * z))));
	} else {
		tmp = x * (2.0 / 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (abs(t) <= 2.864484426612483e-41)
		tmp = Float64(x * Float64(cos(Float64(0.0625 * Float64(b * abs(t)))) * sin(fma(pi, 0.5, Float64(Float64(Float64(0.0625 * abs(t)) * fma(2.0, y, 1.0)) * z)))));
	else
		tmp = Float64(x * Float64(2.0 / 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[Abs[t], $MachinePrecision], 2.864484426612483e-41], N[(x * N[(N[Cos[N[(0.0625 * N[(b * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(Pi * 0.5 + N[(N[(N[(0.0625 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / 2.0), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x, y, z, t, a, b):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf],
	t in [-inf, +inf],
	a in [-inf, +inf],
	b in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z, t, a, b: real): real =
	LET tmp = IF ((abs(t)) <= (28644844266124828983549621919316975406196140503556268027475564542582254570688733394112242861412399625060942742749148237635381519794464111328125e-183)) THEN (x * ((cos(((625e-4) * (b * (abs(t)))))) * (sin((((4 * atan(1)) * (5e-1)) + ((((625e-4) * (abs(t))) * (((2) * y) + (1))) * z)))))) ELSE (x * ((2) / (2))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if t < 2.8644844266124829e-41

   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. Taylor expanded in a around 0

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

   4. Applied rewrites 27.7%

      \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   6. Applied rewrites 27.7%

      \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \left(\left(0.0625 \cdot t\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot z\right)\right)\right) \]

   if 2.8644844266124829e-41 < t

   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. Taylor expanded in a around 0

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

   4. Applied rewrites 27.7%

      \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

   6. Applied rewrites 26.7%

      \[\leadsto x \cdot \frac{\sin \left(\mathsf{fma}\left(\pi, 0.5, \left(\left(0.0625 \cdot t\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot z\right) - \left(b \cdot t\right) \cdot 0.0625\right) + \sin \left(\mathsf{fma}\left(\pi, 0.5, \left(\left(0.0625 \cdot t\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot z\right) + \left(b \cdot t\right) \cdot 0.0625\right)}{2} \]

   7. Taylor expanded in t around 0

      \[\leadsto x \cdot \frac{2}{2} \]
   8. Step-by-step derivation

   9. Applied rewrites 30.3%

      \[\leadsto x \cdot \frac{2}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 30.3% accurate, 13.5× speedup

Math

\[x \cdot \frac{2}{2} \]

FPCore

(FPCore (x y z t a b)
  :precision binary64
  :pre TRUE
  (* x (/ 2.0 2.0)))

C

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

Fortran

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 * (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 * (2.0 / 2.0);
}

Python

def code(x, y, z, t, a, b):
	return x * (2.0 / 2.0)

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[(2.0 / 2.0), $MachinePrecision]), $MachinePrecision]

ReFlow

f(x, y, z, t, a, b):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf],
	t in [-inf, +inf],
	a in [-inf, +inf],
	b in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z, t, a, b: real): real =
	x * ((2) / (2))
END code

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. Taylor expanded in a around 0

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

4. Applied rewrites 27.7%

   \[\leadsto x \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)\right)\right)\right) \]

6. Applied rewrites 26.7%

   \[\leadsto x \cdot \frac{\sin \left(\mathsf{fma}\left(\pi, 0.5, \left(\left(0.0625 \cdot t\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot z\right) - \left(b \cdot t\right) \cdot 0.0625\right) + \sin \left(\mathsf{fma}\left(\pi, 0.5, \left(\left(0.0625 \cdot t\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot z\right) + \left(b \cdot t\right) \cdot 0.0625\right)}{2} \]

7. Taylor expanded in t around 0

   \[\leadsto x \cdot \frac{2}{2} \]
8. Step-by-step derivation

9. Applied rewrites 30.3%

   \[\leadsto x \cdot \frac{2}{2} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(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))))