Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 92.5% → 97.9%

Time: 9.2s

Alternatives: 9

Speedup: 1.1×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x + \frac{\left(y - x\right) \cdot z}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ x (/ (* (- y x) z) t)))

C

double code(double x, double y, double z, double t) {
	return x + (((y - x) * z) / t);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + (((y - x) * z) / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + (((y - x) * z) / t);
}

Python

def code(x, y, z, t):
	return x + (((y - x) * z) / t)

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(Float64(y - x) * z) / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + (((y - x) * z) / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Initial Program: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left(y - x\right) \cdot z}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ x (/ (* (- y x) z) t)))

C

double code(double x, double y, double z, double t) {
	return x + (((y - x) * z) / t);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + (((y - x) * z) / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + (((y - x) * z) / t);
}

Python

def code(x, y, z, t):
	return x + (((y - x) * z) / t)

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(Float64(y - x) * z) / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + (((y - x) * z) / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ x + \frac{y - x}{\frac{t}{z}} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ x (/ (- y x) (/ t z))))

C

double code(double x, double y, double z, double t) {
	return x + ((y - x) / (t / z));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - x) / (t / z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + ((y - x) / (t / z));
}

Python

def code(x, y, z, t):
	return x + ((y - x) / (t / z))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y - x) / Float64(t / z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + ((y - x) / (t / z));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.1%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto x + \frac{\color{blue}{\left(y - x\right)} \cdot z}{t} \]

   2. lift-*.f64 N/A

      \[\leadsto x + \frac{\color{blue}{\left(y - x\right) \cdot z}}{t} \]

   3. remove-double-neg N/A

      \[\leadsto x + \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - x\right) \cdot z\right)\right)\right)}}{t} \]

   4. remove-double-neg N/A

      \[\leadsto x + \frac{\color{blue}{\left(y - x\right) \cdot z}}{t} \]

   5. lift-*.f64 N/A

      \[\leadsto x + \frac{\color{blue}{\left(y - x\right) \cdot z}}{t} \]

   6. associate-/l* N/A

      \[\leadsto x + \color{blue}{\left(y - x\right) \cdot \frac{z}{t}} \]

   7. clear-num N/A

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

   8. un-div-inv N/A

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]

   9. lower-/.f64 N/A

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]

   10. lower-/.f64 98.5

       \[\leadsto x + \frac{y - x}{\color{blue}{\frac{t}{z}}} \]

4. Applied rewrites 98.5%

   \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]
5. Add Preprocessing

Alternative 2: 86.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ t_2 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+305}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) z) t))) (t_2 (* z (/ (- y x) t))))
   (if (<= t_1 -4e+305) t_2 (if (<= t_1 5e+300) (+ x (/ (* y z) t)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (((y - x) * z) / t);
	double t_2 = z * ((y - x) / t);
	double tmp;
	if (t_1 <= -4e+305) {
		tmp = t_2;
	} else if (t_1 <= 5e+300) {
		tmp = x + ((y * z) / t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x + (((y - x) * z) / t)
    t_2 = z * ((y - x) / t)
    if (t_1 <= (-4d+305)) then
        tmp = t_2
    else if (t_1 <= 5d+300) then
        tmp = x + ((y * z) / t)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + (((y - x) * z) / t);
	double t_2 = z * ((y - x) / t);
	double tmp;
	if (t_1 <= -4e+305) {
		tmp = t_2;
	} else if (t_1 <= 5e+300) {
		tmp = x + ((y * z) / t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (((y - x) * z) / t)
	t_2 = z * ((y - x) / t)
	tmp = 0
	if t_1 <= -4e+305:
		tmp = t_2
	elif t_1 <= 5e+300:
		tmp = x + ((y * z) / t)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) * z) / t))
	t_2 = Float64(z * Float64(Float64(y - x) / t))
	tmp = 0.0
	if (t_1 <= -4e+305)
		tmp = t_2;
	elseif (t_1 <= 5e+300)
		tmp = Float64(x + Float64(Float64(y * z) / t));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + (((y - x) * z) / t);
	t_2 = z * ((y - x) / t);
	tmp = 0.0;
	if (t_1 <= -4e+305)
		tmp = t_2;
	elseif (t_1 <= 5e+300)
		tmp = x + ((y * z) / t);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+305], t$95$2, If[LessEqual[t$95$1, 5e+300], N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -3.9999999999999998e305 or 5.00000000000000026e300 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

   1. Initial program 87.6%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{y}{t} - \frac{x}{t}\right)} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{t}} \]

      4. lower--.f64 93.1

         \[\leadsto z \cdot \frac{\color{blue}{y - x}}{t} \]

   5. Applied rewrites 93.1%

      \[\leadsto \color{blue}{z \cdot \frac{y - x}{t}} \]

   if -3.9999999999999998e305 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 5.00000000000000026e300

   1. Initial program 98.7%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]
   4. Step-by-step derivation

      1. lower-*.f64 87.0

         \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]

   5. Applied rewrites 87.0%

      \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -9.2e-16)
   (fma (/ y t) z x)
   (if (<= t 7.2e-106) (/ (* (- y x) z) t) (fma (/ z t) y x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9.2e-16) {
		tmp = fma((y / t), z, x);
	} else if (t <= 7.2e-106) {
		tmp = ((y - x) * z) / t;
	} else {
		tmp = fma((z / t), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -9.2e-16)
		tmp = fma(Float64(y / t), z, x);
	elseif (t <= 7.2e-106)
		tmp = Float64(Float64(Float64(y - x) * z) / t);
	else
		tmp = fma(Float64(z / t), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -9.2e-16], N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[t, 7.2e-106], N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.1999999999999996e-16

   1. Initial program 91.6%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]
   4. Step-by-step derivation

      1. lower-*.f64 85.6

         \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]

   5. Applied rewrites 85.6%

      \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]

      2. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{t} + x} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{t}} + x \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot z}}{t} + x \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{y}{t} \cdot z} + x \]

      7. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{t}} \cdot z + x \]

      8. lower-fma.f64 89.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)} \]

   7. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)} \]

   if -9.1999999999999996e-16 < t < 7.20000000000000025e-106

   1. Initial program 98.0%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - x\right)} \cdot z}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - x\right) \cdot z}}{t} \]

      3. remove-double-neg N/A

         \[\leadsto x + \frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - x\right) \cdot z\right)\right)\right)}}{t} \]

      4. remove-double-neg N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - x\right) \cdot z}}{t} \]

      5. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - x\right) \cdot z}}{t} \]

      6. associate-/l* N/A

         \[\leadsto x + \color{blue}{\left(y - x\right) \cdot \frac{z}{t}} \]

      7. clear-num N/A

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

      8. un-div-inv N/A

         \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]

      9. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]

      10. lower-/.f64 97.3

          \[\leadsto x + \frac{y - x}{\color{blue}{\frac{t}{z}}} \]

   4. Applied rewrites 97.3%

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{t} \]

      3. lower--.f64 88.6

         \[\leadsto \frac{z \cdot \color{blue}{\left(y - x\right)}}{t} \]

   7. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{t}} \]

   if 7.20000000000000025e-106 < t

   1. Initial program 94.2%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]
   4. Step-by-step derivation

      1. lower-*.f64 89.1

         \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]

   5. Applied rewrites 89.1%

      \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]

      2. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{t} + x} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{t}} + x \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot z}}{t} + x \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z}{t}} + x \]

      7. lift-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{z}{t}} + x \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{t} \cdot y} + x \]

      9. lower-fma.f64 91.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)} \]

   7. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (/ (- y x) t))))
   (if (<= z -9.5e+101) t_1 (if (<= z 3.5e+179) (fma (/ z t) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * ((y - x) / t);
	double tmp;
	if (z <= -9.5e+101) {
		tmp = t_1;
	} else if (z <= 3.5e+179) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(Float64(y - x) / t))
	tmp = 0.0
	if (z <= -9.5e+101)
		tmp = t_1;
	elseif (z <= 3.5e+179)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.5e+101], t$95$1, If[LessEqual[z, 3.5e+179], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.49999999999999947e101 or 3.50000000000000015e179 < z

   1. Initial program 92.1%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{y}{t} - \frac{x}{t}\right)} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{t}} \]

      4. lower--.f64 91.3

         \[\leadsto z \cdot \frac{\color{blue}{y - x}}{t} \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{z \cdot \frac{y - x}{t}} \]

   if -9.49999999999999947e101 < z < 3.50000000000000015e179

   1. Initial program 96.3%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]
   4. Step-by-step derivation

      1. lower-*.f64 86.7

         \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]

   5. Applied rewrites 86.7%

      \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]

      2. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{t} + x} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{t}} + x \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot z}}{t} + x \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z}{t}} + x \]

      7. lift-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{z}{t}} + x \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{t} \cdot y} + x \]

      9. lower-fma.f64 87.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)} \]

   7. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{-111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-102}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= t -1.15e-111) t_1 (if (<= t 2.55e-102) (* y (/ z t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / t), z, x);
	double tmp;
	if (t <= -1.15e-111) {
		tmp = t_1;
	} else if (t <= 2.55e-102) {
		tmp = y * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / t), z, x)
	tmp = 0.0
	if (t <= -1.15e-111)
		tmp = t_1;
	elseif (t <= 2.55e-102)
		tmp = Float64(y * Float64(z / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[t, -1.15e-111], t$95$1, If[LessEqual[t, 2.55e-102], N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.15e-111 or 2.55e-102 < t

   1. Initial program 93.9%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]
   4. Step-by-step derivation

      1. lower-*.f64 84.6

         \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]

   5. Applied rewrites 84.6%

      \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]

      2. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{t} + x} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot z}{t}} + x \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot z}}{t} + x \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{y}{t} \cdot z} + x \]

      7. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{t}} \cdot z + x \]

      8. lower-fma.f64 86.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)} \]

   7. Applied rewrites 86.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z, x\right)} \]

   if -1.15e-111 < t < 2.55e-102

   1. Initial program 97.6%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot y}}{t} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y}{t}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{y}{t}} \]

      4. lower-/.f64 53.4

         \[\leadsto z \cdot \color{blue}{\frac{y}{t}} \]

   5. Applied rewrites 53.4%

      \[\leadsto \color{blue}{z \cdot \frac{y}{t}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{z \cdot y}{t}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot z}}{t} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]

      4. lift-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{z}{t}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z}{t} \cdot y} \]

      6. lower-*.f64 67.0

         \[\leadsto \color{blue}{\frac{z}{t} \cdot y} \]

   7. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\frac{z}{t} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-102}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{z}{t}, y - x, x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma (/ z t) (- y x) x))

C

double code(double x, double y, double z, double t) {
	return fma((z / t), (y - x), x);
}

Julia

function code(x, y, z, t)
	return fma(Float64(z / t), Float64(y - x), x)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 95.1%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto x + \frac{\color{blue}{\left(y - x\right)} \cdot z}{t} \]

   2. lift-*.f64 N/A

      \[\leadsto x + \frac{\color{blue}{\left(y - x\right) \cdot z}}{t} \]

   3. lift-/.f64 N/A

      \[\leadsto x + \color{blue}{\frac{\left(y - x\right) \cdot z}{t}} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t} + x} \]

   5. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot z}{t}} + x \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{t} + x \]

   7. associate-/l* N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{t}} + x \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} + x \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)} \]

   10. lower-/.f64 98.4

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{t}}, y - x, x\right) \]

4. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)} \]
5. Add Preprocessing

Alternative 7: 77.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{z}{t}, y, x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma (/ z t) y x))

C

double code(double x, double y, double z, double t) {
	return fma((z / t), y, x);
}

Julia

function code(x, y, z, t)
	return fma(Float64(z / t), y, x)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]

Derivation

1. Initial program 95.1%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]
4. Step-by-step derivation

   1. lower-*.f64 79.7

      \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]

5. Applied rewrites 79.7%

   \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]

   2. lift-/.f64 N/A

      \[\leadsto x + \color{blue}{\frac{y \cdot z}{t}} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\frac{y \cdot z}{t} + x} \]

   4. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y \cdot z}{t}} + x \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{y \cdot z}}{t} + x \]

   6. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{z}{t}} + x \]

   7. lift-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{z}{t}} + x \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\frac{z}{t} \cdot y} + x \]

   9. lower-fma.f64 81.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)} \]

7. Applied rewrites 81.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)} \]
8. Add Preprocessing

Alternative 8: 40.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ y \cdot \frac{z}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* y (/ z t)))

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y * (z / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return y * (z / t);
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.1%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{z \cdot y}}{t} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{z \cdot \frac{y}{t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{z \cdot \frac{y}{t}} \]

   4. lower-/.f64 37.4

      \[\leadsto z \cdot \color{blue}{\frac{y}{t}} \]

5. Applied rewrites 37.4%

   \[\leadsto \color{blue}{z \cdot \frac{y}{t}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{z \cdot y}{t}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{y \cdot z}}{t} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]

   4. lift-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{z}{t}} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\frac{z}{t} \cdot y} \]

   6. lower-*.f64 42.2

      \[\leadsto \color{blue}{\frac{z}{t} \cdot y} \]

7. Applied rewrites 42.2%

   \[\leadsto \color{blue}{\frac{z}{t} \cdot y} \]

8. Final simplification 42.2%

   \[\leadsto y \cdot \frac{z}{t} \]
9. Add Preprocessing

Alternative 9: 37.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ z \cdot \frac{y}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* z (/ y t)))

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = z * (y / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return z * (y / t);
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.1%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{z \cdot y}}{t} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{z \cdot \frac{y}{t}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{z \cdot \frac{y}{t}} \]

   4. lower-/.f64 37.4

      \[\leadsto z \cdot \color{blue}{\frac{y}{t}} \]

5. Applied rewrites 37.4%

   \[\leadsto \color{blue}{z \cdot \frac{y}{t}} \]
6. Add Preprocessing

Developer Target 1: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (< x -9.025511195533005e-135)
   (- x (* (/ z t) (- x y)))
   (if (< x 4.275032163700715e-250)
     (+ x (* (/ (- y x) t) z))
     (+ x (/ (- y x) (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x < (-9.025511195533005d-135)) then
        tmp = x - ((z / t) * (x - y))
    else if (x < 4.275032163700715d-250) then
        tmp = x + (((y - x) / t) * z)
    else
        tmp = x + ((y - x) / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x < -9.025511195533005e-135:
		tmp = x - ((z / t) * (x - y))
	elif x < 4.275032163700715e-250:
		tmp = x + (((y - x) / t) * z)
	else:
		tmp = x + ((y - x) / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x < -9.025511195533005e-135)
		tmp = Float64(x - Float64(Float64(z / t) * Float64(x - y)));
	elseif (x < 4.275032163700715e-250)
		tmp = Float64(x + Float64(Float64(Float64(y - x) / t) * z));
	else
		tmp = Float64(x + Float64(Float64(y - x) / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x < -9.025511195533005e-135)
		tmp = x - ((z / t) * (x - y));
	elseif (x < 4.275032163700715e-250)
		tmp = x + (((y - x) / t) * z);
	else
		tmp = x + ((y - x) / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Less[x, -9.025511195533005e-135], N[(x - N[(N[(z / t), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[x, 4.275032163700715e-250], N[(x + N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -1805102239106601/200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- x (* (/ z t) (- x y))) (if (< x 855006432740143/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z))))))

  (+ x (/ (* (- y x) z) t)))